Up next in 10
Operations With Polynomials
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Mar 2, 2025
We're officially starting chapter 5 today and in the first lesson, we'll learn how to multiply, divide, simplify monomials and expressions involving powers. We'll also learn to add, subtract and multiply polynomials.
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0:00
a pretty um easy section five for one
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we're going to talk
0:07
[Applause]
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about operations with like uh pols right
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the polinomial we talked about that in
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like very first here right here we
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talked about it like earlier on the year
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I think second quarter right poal is
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like anything in the form ax² plus BX
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plus C and all of that stuff but first
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we need to learn how to do operations
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with monomials a monomial is an
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expression that looks like this a x the
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n a x the m y the N it's just one one
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term right one term so that's called a
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monomial it's just one term okay so now
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what are we doing here we're going to
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learn how to uh the product rule with
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monomial how do you multiply them right
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if I have for example X2 * x 5 what is
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that give me that's the kind of stuff
1:01
that we talking about in the very first
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uh Sol section here or Sol header or
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whatever right if I have for example X
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the^ of a and I want to multiply that
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times x^ of B what I'm actually doing is
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this because the base is the same right
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all you're going to do is just add the
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exponent does that make sense right we
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have the same base the base is X and X
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right this is X to a * X to B if because
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the base is the same the Bas is the X
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all you do is just make it one single
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expression and then add the
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exponent example 3 4 * 3 6 is = 3 4 + 6
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which is 3
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10 right you guys did that before yeah
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all right all so it's like I desperately
1:51
needed this reminder though all right so
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so that's the product R for monomial
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right so x 5 * x 6 is x 11 as long as
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the base is the same you add the
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exponent right now this is just a quick
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we're just going over the rules real
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quick and then we're going to get into
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the meat of the the thing right now when
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you divide them now division is what
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really throws a lot of people off people
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get confused when it comes to division
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right for example if I have x to the A
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and I'm dividing by x to the B what I'm
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going to do is it becomes again the base
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is the same is X the a minus B right
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example
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5^ 7 over 5 3r is = 5^ 7 - 3 does that
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make sense one which is 5 to 4 no right
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5 to four then you can find what 5 to
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four
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isent so when yeah when exponent Z which
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makes it one over one that's right right
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so x a x b is X a minus B this is the
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one that you move you move move this up
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here and you subtract it because every
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time you switch from the bottom up or
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top to down you change the the sign of
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the exponent right so extra a extra B is
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extra a minus B so 537 over 5 3 is 5 7 -
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2 which is 5 4 but you have to make sure
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the base is the same if the base is not
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the same this is the base you cannot do
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that right for example if I have 6 7
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over 5 to 3 is not going to be 6 7 - 3
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you can't do that because the base is
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not the same you have to make sure the
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Bas is the same right are we
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understanding this all right it seems
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like everybody's on the same page so we
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good right and now we also going to deal
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with negative exponents so how do I deal
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with a negative exponent so if I have x
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n right that's no
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I right x n isal to 1 x n
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this is again one of those formulas that
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you need to remember right it's not a
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formula it's just a a rule right so
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example we have 7^ -2 it becomes 1 7
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2 right x n is 1 X N example 7 -2 is =
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to 1 7 2 which is basically 1 49 right
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so what if I have
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84 sh what what would I give me what 8
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to4 if I want to get rid ofx what what
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would I give me according to this rule
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oh I don't know 1
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8 1 8 4 right this is 1 over 8 to 4 yeah
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dividing one to finish the equation will
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we do the five four to finish it well it
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depends on the what question is is the
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instruction is asking you to do it then
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yes but if not then you just leave it
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here you point off do it anyway
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I don't know it all Bas it's based on
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the question that is being asked if I
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ask you to go further then you can now
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here I know that 7 squ is 49 so I'm not
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going to leave it as 1 7 squ it's 1 49
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obviously right okay uh tell me what is
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6 to the power of
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-7 if you want to get rid of negative
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exponent what would I give you 1 6 are
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you tell me
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she saved you did you know it I'm sorry
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I didn't hear your all good all right
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the next one is called the power of a
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power right power of a power so X the a
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raised to the power of b equals x a what
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right X the a the whole thing raised to
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the power of B is equal to same base to
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the power of ab all these are just rules
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right the rules that we going over so
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example 3 7 ra^ 2 is equal 3 14 right
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some people
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confuse this and that sometimes a lot of
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people have done that I've seen this
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like my years before right now look
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there's a distinction between this here
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and then this here right A lot of people
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who see this extra a extra B and they go
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oh this is extra a * B no when the base
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is s you multiply and then what do you
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do to the exponent plus right you add
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and then here you multiply so you have
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to remember that this two things are
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different right these two things are
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different a lot of people have done I'm
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telling you like I've seen it so many
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last year I saw it right so I've seen it
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a lot of times so example 3 7 2 is 3 7 *
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2 which is 3 power
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14 is anybody lost just overlo now
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it's just too many remember a little F
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it's just so many because you know why
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I'm going fast because it's just rules
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it's in your book right oh sure yeah
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thanks for reminding me it come never
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ask about homework I will no the next
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one is called the power of a product
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right x x y ra the power n isal to x n *
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x to the N example if I have X Y ra to
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the 7
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each uh expression here is going to be
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ra^ of 7 will be x 7 * y 7 okay so this
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this is called the uh the power of a
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product
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okay that's called The Power of a
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product and then we also
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have the power of the quotient right
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quotient so have X Y to the a ra to the
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a each one of the expression is going to
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be ra^ a so X Y the holding ra power a
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is equal to x a over y the a can we
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leave it at that or we solve it further
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well you can't do anything here okay
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obviously right now we have 2 over 5^ 5
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that is 2 5th over 5 5th right because
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again each one of them is going to be ra
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to the power 5 right now the next one
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here do you see what I did is I switch
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the the exponent to a negative right so
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when the exponent is switch to a
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negative all you do is flip the the
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expression that's inside and then change
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the the exponent to a
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positive um integer right or real number
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whatever the thing is so for example X
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over
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y^8 is going to be y x power of a I give
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you an example here 53 7 3
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76 it becomes 7 3 to 6 and then each one
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will be raised to the power 6 so 7 6 so
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you just fli your X and Y you just flip
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your X and Y and changes into a posi
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exponent right these are just the basic
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rules that are going to dictate how
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we're going to um how we're going to
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deal with this section it's almost like
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this is your baby stage first you crawl
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then you walk are we in the crawling
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stage right now crawling first so it's
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about to get really hard no no is to
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craw walk and cry right now zero power I
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love this one right any number raised to
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the power of zero is always going to
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give you what one right no matter what
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10,000 to 0 is 1 5,000 to 0 is
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one one yeah not one as y u a n but one
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as o n e
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all right y u n y u a
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n all right can I erase this here may I
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erase
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it you said no why yes
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okay excuse
10:27
me does exist it's all real numberb it
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does no don't worry
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like
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actually all right now we're going to
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learn how to do some quick
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simplification here simplifying an
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expression right so let me give you an
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example so now we're going to apply the
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rule that we just learned if I have 2 a
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right ra to^
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-2 time 3
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a right you just
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sub B and then b
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s and then
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C to2 right and they want us to simplify
11:17
this expression right so we need to
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simplify this so I'm looking at I'm like
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all I can simplify this right so what I
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want to do I want to rearrange
11:29
everything so I'm going to put almost
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like it's almost like putting like terms
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together right so I have numbers and I
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also have variables right so what I'm
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going to do is this I'm going to do two
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right times what three right 2 * 3 cor
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right wait so can you take them out of
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there I you can take them out of the
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parentheses okay right so not like a
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whole bunch yeah okay and then now look
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I have 8 -2 * what 8 3 you see what I'm
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doing here right and and I have b^ 2 * C
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-2 so I'm going to rearrange it so that
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way it looks easier to deal
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with why is your head down
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Mary CU my head hurt all right keep it
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up if you lift your head off like they
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won't hurt trust
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me I'm
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serious no no like pick your eyes gaze
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upon the board and you'll be good 2 * 3
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is equal to what
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six
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six I hope you knew that now here I have
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the same base right
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8 so what is that going to give me 8 to
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1 or 8 to the first right because it's
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-2 + 3 which is equal to 1 so this is a
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right now B squ is by itself so I'm
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going to leave it alone now here's the
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thing again this is one of those
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fundamental rule that we're going to do
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in this chapter we can never keep a
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negative exponent if you ever have a
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negative exponent and you turn it in
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like that it's I'm going to do do some
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point right so what I do with this it's
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a negative yeah c 1/ c s so that becomes
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1 / c^ 2 right and don't leave it like
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this now I can simplify that c is going
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to be in the bottom so it's going to be
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6 A and B squ and this is your final
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answer right does that make sense cuz we
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are we are multiplying the top by
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one right so I give a 6 a² and then we
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going to change this into a fraction
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because we learned that earlier this is
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called a negative exponent rule right
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all right this supposed to be foiled no
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we it's not going to foil this right so
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now another one now let's say I
13:48
have now stop me there's something that
13:50
you don't understand so I can go over it
13:52
again right cuz I do not want
13:55
to I do not want to leave any stones on
13:58
turn
14:03
oh no
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that's yeah you said it was going to be
14:09
like two homk assments a
14:11
week was it that's what you said that's
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what I
14:15
said really yeah if there's one thing
14:18
you can learn about humans is that they
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change they they Li
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they I'm I'm I'm like you know from G
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Galan right
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I'm I'm fle so I I can change right I'm
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not God right so so you do lie I don't
14:38
lie I things can change you said you
14:41
said it's going to be it might things
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can change I can change my mind that's
14:47
what makes me I can change my mind if I
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want mind I'm allowed to right hold up
14:53
so now we doing this here right now
14:56
watch
14:57
so if so look right wait a minute we
15:01
have we have that right so we're going
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to have to simplify this we we need to
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simplify this right so I have X so
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here's what I'm going to do here x² X is
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7 right so how do I simplify that into
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one single variable with one exponent
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what is that going to give me sub
15:19
subtract right we're going to subtract 2
15:22
-
15:23
7 which is what5 so you should be X to5
15:27
here does it have to be x with X and Y
15:30
and Y that's y excuse me do you have to
15:34
do like X and X and Y yeah you got to do
15:38
X and Y no cuz they don't have the same
15:41
base they have to have the same base
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right so y got NE 5 here I did what two
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right - 7 that's what we did in the last
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rule that we talked about this is called
15:53
the negative exponent right the quo rule
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right you're going to do 2 - 7 is5 and
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then here you're going to what 4 minus
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what 3 right so this is 4 - 3 so that is
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y power of one and we have nothing under
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here right but now remember we cannot
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have a negative
16:13
exponent so what is that going to give
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me
16:16
then it's going to give 1 5 there you go
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thank you very much y * 1 x 5 and now I
16:24
can just multiply Y and 1 so that give
16:26
you what y over what
16:29
x to the fifth right and like I said my
16:32
f saying this is like a walk in the park
16:35
walk in the park
16:37
this yeah this one actually be good yeah
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I told you
16:42
all I told you all it was going to be
16:48
fun that was fun
16:51
fun all right next let's say we have
16:54
this here right next now -2 a
16:59
it's going to get interesting now four
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right
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over uh B to the two and the whole thing
17:07
ra to the power of three right so this
17:09
is called a power of a power right so
17:12
I'm going to I'm going to raise each one
17:14
of those entries of numbers and
17:17
variables to the power of three right so
17:19
the first one's going to give you -2 to
17:22
the power of what three and then a is
17:25
going to be raised to the power of what
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remember seven no that's seven ah
17:29
12 you do what 4 * 3
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right this is where you have to be
17:34
extremely careful and then B is going to
17:35
be to the power of what 6 cuz 2 * 3 so B
17:39
to
17:40
6 is anybody confused here wait can you
17:43
just say it again so we're going to
17:45
start with each one of those uh numbers
17:48
right and variable so -2 is the^ three
17:51
because this is 2 to the one right and
17:53
then a because this is four you do 4 * 3
17:57
that gives you 8 to 12 and and then B is
17:59
square and it's also
18:01
power6 right and then now add the three
18:05
to everybody yeah every sing one of them
18:07
multiply right and now -2 to 3 is what
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12 no -2 to 4 3 is8 right so this
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812 V6 right and then that's it we stop
18:20
here all right there's nothing else to
18:22
do here we stop in this this is the
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final all right last one and then we get
18:28
no no no
18:29
finished we are not
18:31
finished all right now what
18:34
you your head
18:43
hurt a question
18:46
yes
18:48
yes need oh okay all right now watch
18:54
here so let me ask you a question how
18:57
would you solve this here a to3 flip it
19:01
flip it thank you and what do we get 4
19:05
a that's it very good
19:10
right so 4 cube is what 64 over what the
19:15
thir thank you very much so you getting
19:18
it what is it like huh what's it like
19:22
what to walk in the park oh yeah it's
19:24
like a walk in the
19:26
park walk in the park
19:29
all right so the next thing we going to
19:31
learn next oh yeah we got we got a
19:34
couple more things here come
19:46
on this
19:49
guy this
19:51
guy took
19:55
years you can't act casual if you don't
19:57
get the sound stop
20:10
all
20:26
right is it by
20:38
wolver Jack press the button press the
20:41
button I'm not going to press the button
20:43
press the button
20:48
Buton I'm going to beat them all all
20:50
right so now we're going to talk about
20:52
the degree of a polom right how do we
20:55
find the degree of a polom so that's
20:57
what we're going to discuss
21:02
Tom that the big what's the second word
21:06
after
21:08
big the big best
21:14
degree so let's say we have ax²
21:18
right plus BX plus
21:20
C right and I want to find the
21:23
degree of this polinomial so basically
21:26
the degree of the polinomial is by that
21:28
definition the degree of the monomial
21:32
with the highest degree right how many
21:34
monomial do I have
21:36
here three right I have three monomial
21:38
so I have this guy I have this guy and I
21:41
have this guy right so if I want to find
21:45
the degree of this entire polom it's
21:47
going to be the degree of the highest of
21:50
the mon monomial with the highest degree
21:52
right so the degree is this here this is
21:54
square this is one and this is zero so
21:57
what's the highest here two two is the
22:00
highest degree right cuz I'm looking at
22:02
the exponent the exponent is your degree
22:04
so therefore I'm going to say the degree
22:06
of this polom is what is two is it that
22:08
easy that easy this why you know like I
22:12
don't know why everybody complains cuz
22:14
it's that easy now I'm going to give you
22:16
some examples and we're going to try and
22:17
figure it out right so let's start with
22:19
the easy ones so let's say we have x 7 +
22:24
3x CU -
22:27
1 x seven so what's the degree of this
22:30
polinomial s seven because what's the
22:33
highest monomial here the degree the
22:35
monomial the highest degree seven does
22:38
everybody understand that yes right
22:41
excuse me no okay so this is called a
22:44
polom right because you have poly means
22:47
many right plural right so this polom
22:51
has how many monomials three right 1 2
22:54
and three correct so now to find the
22:57
degree of this monomial what you do is
22:59
this you find the monomial with the
23:02
highest degree and because he has the
23:04
highest degree that becomes by the F the
23:08
degree of the the entire polinomial
23:10
right so the degree is the exponent here
23:13
right so if you look we have ax² + BX
23:15
plus C so if I look at all these three
23:18
monomials which one has the highest
23:21
exponent so therefore the degree of this
23:23
polom will be two right now here we have
23:27
X7 + 3x - 1 so what will be the degree
23:30
of this polinomial here seven because
23:33
the monomial with the highest degree is
23:35
seven right now the funny thing is this
23:38
with this section there are some some
23:40
some differences we have some uh
23:49
uh uh what was I going to
23:56
see do we have homework today
23:59
excuse me no we only have 15 more
24:01
minutes yes no 15 all right so white I
24:10
have did you charge for
24:14
late
24:18
$20 all right here
24:21
right who's going to freak
24:25
out oh St the Sprite I'm buring today CU
24:29
I'm yes you will can I just no now
24:32
here's the thing right no it's not five
24:34
either gotcha so what we do here is this
24:38
right here's what we do we still have a
24:40
monomial here right because it's one
24:42
expression the bottom one so we're going
24:43
to add even though the Bas are the same
24:46
we're still going to add this 4 + 3 is
24:48
what seven seven therefore the degree of
24:50
this polinomial is what unit overall
24:53
power so here we know five five is less
24:57
than seven unit what are we do right div
25:02
so
25:04
4 is seven right this is five and this
25:07
is three the highest degree is seven so
25:10
therefore the degree of this polom is
25:12
seven all right now what if I have this
25:16
here one un so
25:21
everything so what if I have this
25:24
here so what do you say
25:28
yes is that first of all we have to look
25:30
at it right well here's a problem you
25:33
give us this huh why wait where did you
25:35
get the square root from square root of
25:37
x right any number square root is that
25:39
number the power of what if I square
25:42
root of x is 12 right anytime you have
25:46
sare root of x hold where' you get the
25:48
square root from square root of x is the
25:50
equivalent of x to the 12 right now the
25:53
problem is this x 12 because this is
25:57
radical this is not a polinomial this
26:00
cannot be called
26:02
not a polom
26:05
because we have a radical right anytime
26:07
you see a square root involved in the
26:11
term in the expression this is not a
26:13
polom it disqualifies it as a polom so
26:16
this is not a polom does that make sense
26:19
right because we have that right because
26:22
there a verticle here is disqualify this
26:24
expression and this expression is not a
26:26
polinomial we don't need to find
26:28
there's no such thing does that make
26:31
sense all right now what if I have this
26:34
here
26:36
um x -3 + 2x -2 -
26:42
1 right what do you see here it's going
26:46
to be the
26:48
2- well
26:50
technically it can't be either because
26:52
guess what we have if I Tang this
26:55
becomes what 1 X Plus what what 2x^2 - 1
27:00
this again is not a polom because this
27:02
is a rational expression right because
27:06
two over two over yeah 2 over x s right
27:09
okay so because this is like a a
27:12
negative exponent it getting
27:14
disqualifies it as a polinomial it
27:15
cannot be a polinomial so there's no
27:17
degree to this okay because those
27:19
numbers have to what positive integers
27:22
right if they're not positive integers
27:24
we do not have a polinomial so therefore
27:27
this is not a polinomial
27:28
so therefore you can't say that so you
27:30
just it disqualifies us does that make
27:32
sense what do you use pols for to graph
27:36
functions in real graph functions and
27:38
then determine what the function is
27:39
going to look like and that's going to
27:41
determine a whole lot of things we just
27:43
went on me and me and Dave me Dave Nate