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# Find the coefficient of x2in the expansion of (1 + 2x)^5(2-x)^6

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I've never done this kind of problem before and my teacher's never taught it, she expects us to have learned it last year even though we didn't. Can I get a full explanation on how to do it?

Feb 14, 2021

#1
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This isn't the best way to do it since it's just brute force but it's still an answer I guess:

I'm sure you've learned how to expand brackets right? If you haven't, you basically expand it like this:

Now, we can expand all of this:

(1+2x)5(2-x)6

--> (1+2x)5 = (1+2x)(1+2x)(1+2x)(1+2x)(1+2x)
= 32x+ 80x4 + 80x3 + 40x2 + 10x + 1

--> (2-x)6 = (2-x)(2-x)(2-x)(2-x)(2-x)(2-x)
= x6 - 12x5 + 60x4 - 160x3 + 240x2 - 192x + 64

So if we put this together:

(1+2x)5(2-x)6  = (32x+ 80x4 + 80x3 + 40x2 + 10x + 1)(x6 - 12x5 + 60x4 - 160x3 + 240x2 - 192x + 64)

Then when you expand this, you get:

32x11 - 304x10 + 1040x9 - 1240x8 - 790x7 + 2537x6 + 76x5 - 2180x4 - 320x3 + 880x2 + 448x + 64

But all you need to pay attention to is 880x2 which means the coefficient of x, in other words your answer, is 880.

Feb 14, 2021
#2
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Sorry I just realized there is a much quicker way to figure this out.

Going back to this step:

(1+2x)5(2-x)6  = (32x5 + 80x4 + 80x3 + 40x2 + 10x + 1)(x6 - 12x+ 60x4 - 160x3 + 240x2 - 192x + 64)

You can find the coefficient of x2 by multiplying the terms that make x2 and adding them up:

(10x)(-192x) + (40x2)(64) + (1)(240x2)

= -1920x2 + 2560x2 + 240x2
= 880x2

Guest Feb 14, 2021
#4
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thank you haha, I sure could DO it this way, however the class I'm in, I know that would actually be incorrect lol. The answers right but the method isn't what would be wanted.

Guest Feb 14, 2021
#3
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Nice solution guest and logarhythm!

Here's an alternate solution:

We need to find the coefficients of x^2, x, and constants of both $$(1+2x)^5$$ and $$(2-x)^6$$.

To do that, use the binomial theorem (explained here: https://www.mathsisfun.com/algebra/binomial-theorem.html )

Knowing that, the constant, x, and x^2 coefficients of $$(1+2x)^5$$ are 5 choose 0 = 1 , 5 choose 1 * 2 = 10, and 5 choose 2 * 2^2 = 40, respectively, and the constant, x, and x^2 coefficients of $$(2-x)^6$$ are 6 choose 0 * 2^6 = 64, 6 choose 1 * -1 * 2^5 = -192, and 6 choose 2 * 2^4 = 240, respectively.

The x^2 term of the product of the 2 expressions will happen when an x term of one expression is multiplied by the x term in the other expression, or when the x^2 term in one expression is multiplied by a constant term in the other expression.

That is equal to $$64\cdot40+-192\cdot10+240\cdot1=\boxed{880}$$

Feb 14, 2021