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### Course: Algebra (all content) > Unit 10

Lesson 28: Proving polynomial identities# Analyzing polynomial identities

The video focuses on the critical analysis of polynomial manipulations, emphasizing the importance of checking each step for errors. It provides practice on identifying valid polynomial identities and highlights the significance of correct algebraic multiplication in solving polynomial equations.

## Want to join the conversation?

- I understand that the second step: (x-2) (x-2) expands into the form: ( x^2 -4x +4 ).

However, if this part of the equation is in the form (a-b) ^2 , which expands into

(a^2 - 2ab +b^2) , why then isn't the middle term positive? If x= a and -2= b , shouldn't it also equal ( (x) ^2 -2 (x) ( -2) + (-2)^2 ). Here one can see that the statement equals x^2 + 4x +4 . Does this mean that the b term only includes 2 and not -2? Thanks.(5 votes)- You forgot something: It's a
*minus*b. So you are*subtracting a positive*,*not a negative*. So`b`

itself**is positive**, so the middle term is negative.(21 votes)

- On the second exercise, first choice, couldn't it be true if either x or y was equal to 0?(7 votes)
- It could be, but for the identity to be valid it must be true in all possible cases.(12 votes)

- They went wrong in the 4th step.When you multiply 2 negative numbers you get a positive number. So -3 x -4x = +12x(3 votes)
- Does anyone here actually know what he's talking about? Cause I'm extremely confused. T-T(4 votes)
- he's just continuing with the last video. The link is https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-factor/x2ec2f6f830c9fb89:poly-identities/v/polynomial-identities-intro(3 votes)

- Hi, sorry I know this question doesn't exactly relate to the video but it is in the section of my textbook called Quadratic Identities...

I'm stuck on this question:

Express m^2 in the form a(m-1)^2+b(m-2)^2+c(m-3)^2

Would really appreciate some help or otherwise direction to a video which could help.

Thank you!(1 vote)- They want you to find a, b, and c that satisfy a(m-1)²+b(m-2)²+c(m-3)²=m²

To start, let's expand the left:

a(m²-2m+1)+b(m²-4m+4)+c(m²-6m+9)=m²

am²-2am+a+bm²-4bm+4b+cm²-6cm+9c=m²

Now regroup it:

am²+bm²+cm²-2am-4bm-6cm+a+4b+9c=m²

(a+b+c)m²-2am-4bm-6cm+a+4b+9c=m²

On the left, we have constants, a+4b+9c. On the right, no constants. So a+4b+9c must equal 0.

Similarly, there are no linear (m¹) terms on the right. So -2am-4bm-6cm=0.

So the m² terms must be equal. (a+b+c)m²=m².

Let's factor the second-to-last equation. We get

-2m(a+2b+3c)=0

So either m=0, or a+2b+3c=0. We reject m=0, because we want this to be true for all m. So a+2b+3c=0.

For the last equation, we subtract m² to get (a+b+c)m²-m²=0

m²(a+b+c-1)=0

So a+b+c-1=0, or a+b+c=1

Now we finally have a system of equations:

a+4b+9c=0

a+2b+3c=0

a+b+c=1

I'll skip the algebra to solve the system, but we get solutions a=3, b= -3, c=1

So m²=3(m-1)²-3(m-2)²+(m-3)²(8 votes)

- In the second problem, how did he get 4n by squaring? I tried it myself and I still could not figure it out. Of course the problem couldn't be solved without the 4n, but how did Sal get to that point?(1 vote)
- (n+2)^2=(n+2)(n+2)

When you multiply (n+2) and (n+2), you get n^2+2n+2n+4. Simplifying, you get n^2+4n+4 since 2n+2n=4. The 4n comes from the middle terms of the expanded polynomial, 2n and 2n, being added together.

Similarly, if we had (x+5)(x+5), the answer would be x^2+5x+5x+25, or x^2+10x+25. The 10x comes from the middle terms, 5x and 5x, being added together.

Hope this helps!(6 votes)

- If x^2+[1/(x^2)]=72 then find the value of x^3+[1/(x^3)](3 votes)
- I understand that the second step: (x-2) (x-2) expands into the form: ( x^2 -4x +4 ).

However, if this part of the equation is in the form (a-b) ^2 , which expands into

(a^2 - 2ab +b^2) , why then isn't the middle term positive? If x= a and -2= b , shouldn't it also equal ( (x) ^2 -2 (x) ( -2) + (-2)^2 ). Here one can see that the statement equals x^2 + 4x +4 . Does this mean that the b term only includes 2 and not -2? Thanks.(2 votes)- If you're using the identity (a-b)²=a²-2ab+b², then b=2 in your expression, not -2. You're subtracting 2, not subtracting -2.(1 vote)

- Can someone help me with this algebra question?

Prove 3(x+y)^2 - (x-y)^2 = 2x^2 + 8xy^2 +2y^2

Thank you so much!(1 vote)- First you have the expression 3(x+y)^2 - (x-y)^2.

3(x+y)^2 - (x-y)^2

= 3(x^2 + 2xy + y^2) - (x^2 - 2xy + y^2)

= 3x^2 + 6xy + 3y^2 - (x^2 - 2xy + y^2)

= 2x^2 + 6xy - (-2xy) + 2y^2

= 2x^2 + 8xy + 2y^2

(In your question, you wrote "8xy^2". I'm guessing you meant "8xy".)(2 votes)

- I'm a little confused about the first example, I thought that (x-2)^2 is equal to (x-2)(x+2), because negative times positive is negative, but negative times negative is positive!? Wouldn't (x-2)(x-2) equal (x+2)^2?(1 vote)
- An exponent is repetitive multiplication of the same value.

For example: 5^2 = 5*5; (-5)^2=(-5)(-5)

The same concept extends to polynomials.

The expression: (x-2)^2 is telling you to multiply the factor (x-2) with itself = (x-2)(x-2)

In your examples, you are trying to change the factor's that the exponent applies to. All your new versions are completely different than the original. You can verify this if you use FOIL to multiply your various options:

(x-2)^2 = (x-2)(x-2) = x^2-2x-2x+4 = x^2-4x+4

(x+2)^2 = (x+2)(x+2) = x^2+2x+2x+4 = x^2+4x+4

(x-2)(x+2) = x^2+2x-2x-4 = x^2-4

Hope this helps.(2 votes)

## Video transcript

- [Voiceover] What I
hope to do in this video is give ourselves some practice at critically looking at how folks manipulate polynomials. And the reason why this
is useful is because it's useful to be able
to do this to yourself as you manipulate polynomials to say wait, what did I exactly do there? Or a lot of times if you're reading a math or a science book they're going to do some proof or something like that, and they're gonna say oh well you know, it's obviously from
this step to this step, and you're gonna try to follow it and you're just like, well
does that make sense? So it's a really useful
muscle to be able to see do these steps, or whoever
manipulated the polynomial, does it make sense to you? And especially if it's
you, it's super useful to be able to find if there are errors, and to correct them, it'll just give you a better critical
eye for this type of thing. So let's just start with this one. We have 4x minus three
times x minus two squared. And it looks like this
person over five steps tries to expand it out. And so what I encourage you to do, pause the video right now, and see if they did it correctly. And if they didn't do it correctly, try to identify on what
step they messed up. All right, so assuming you had a go at it, let's do this together. So as we go from the first expression to the second, to step one, what do they do? Well, they just expanded
out x minus two squared. X minus two squared is just x minus two times x minus two, they haven't done anything to the 4x minus three yet. Now what do they do in this step, so that seems correct. So in step two, it looks
like they're just trying to multiply x minus two times x minus two. So you have x times x,
which would be x squared. You have x times negative two, which would be negative 2x. You have negative two times x, which would be negative 2x. Then you have negative
two times negative two, which would be positive four. So it looks like they
multiplied this out correctly, so step two we're still doing good. All right, now what do
they do in step three? And this whole time, 4x minus three, they haven't really touched it yet. So they're just trying to simplify it, and all they did is they
added these two middle terms. Minus negative 2x minus 2x is going to be negative 4x, so this still, this still looks correct. The x squared didn't change,
the plus four didn't change, they just added these middle two terms. Now as we go to this next step four, well now they're trying to multiply these two expressions,
so they're doing some algebraic multiplication. So let's see if we can figure this out. So we have 4x times, let me do this in a new color, I'm getting bored of that magenta. All right, so we have, we have 4x times x squared, which is indeed 4x to the third power. Then you have 4x times negative 4x, which is gonna be negative 16x squared, so they did that right. Then you have 4x times four,
which is going to be 16x, and they wrote that right over there. Then, you're going to have negative three times x squared, which
is negative 3x squared, we see that right over there. Then you're going to have negative three times negative 4x, which is going to be positive 12x, and they wrote negative 12x. So they forgot, they saw
a negative, negative, but they still put a negative there. Negative three times negative 4x, a negative times a negative
is gonna be a positive. Positive 12x. So they made an error here, and then they said negative three
times positive four, which is indeed negative
12, so this part is right. So the error, this thing should read positive 12x. So the error they made is in step four. Step four is the error,
and then that ended up giving them the wrong answer here, because they did a minus 12x instead of, if this was a negative
12x, then negative 12x plus 16x got you this 4x. But we know it's supposed to be plus 12x. So it really should be 28. This should be 28x right over here. But they really messed up. If you take that error,
they did this step right. But step four is where they
actually made the error. So let's keep going, let's give ourselves a little bit more practice at looking at ways to manipulate polynomials and see if they're valid. So here, this comes from an
exercise on Kahn Academy. Let's see which of these
are valid identities. Which of these are valid statements? So this first one, 2x plus y, times 4x, 2x plus y times 4x minus 2y is all of this business right over here. Le'ts just multiply it out. 2x times 4x is going to be 8x squared. 2x times negative 2y is going to be negative 4xy. And then, let me switch colors, y times 4x is going to be plus 4xy. And then y times negative 2y is going to be minus 2y squared. And so let's see, these
two, did I do that right? Let's see, 2x, times negative 2y is negative 4xy. And then I had 4x times y is positive 4xy. So these two are gonna cancel out. So this is already, this is looking shady. So all we're left with is 8x squared minus 2y squared, if we factor a two out, it's going to be two times 4x squared minus y squared. So this is not a true
statement right over here. Now let's try this one. N plus two squared minus n
squared is equal to this. But what's n plus two squared? That's going to be n squared plus plus 4n, it's gonna be 2n plus 2n, it's gonna be plus 4n plus four, and then we're gonna
subtract out an n squared, these cancel, so we're
gonna have 4n plus four, which is equal to four times n plus one. So this one right over here works out. This is a true, I guess in
the language of this question, it is a valid identity, or you could say it's a true statement,
this equation is true. And then we have this
last one right over here. And once again, let's see
if we can multiply it out. Let me go down here into the black space. So if I have a times 2a,
that's going to be 2a squared. And then a times one, is going to be plus a. And then if I have b times 2a, it's going to be plus 2ab, and then finally if I have b times one, it's going to be plus b. And then out here, we are subtracting a b. So over here we're going to subtract a b. These characters are going to cancel out. And then we're left with 2a squared plus a plus 2ab. And it looks like they factor out an a, so let's see if we can
factor out an a ourselves. So if we factor out an a,
we're gonna be left with this first term is going to be 2a, this right over here,
if we factor out an a, is going to be plus one, and
this if we factor out an a is going to be 2b. And that's exactly what
they wrote over here, they just wrote it in a different order. A times 2a, plus 2b plus one. So this is legit. So hopefully that gave
us some good practice at critically looking at whether people are making valid statements. And this is going to be the most useful to figure out if you are
making valid statements. All right, hopefully you enjoyed that.