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Expand the product (x - 2)^2 (x + 2)^3. What is the product of the nonzero coefficients of the resulting expression, including the constant term?

 May 8, 2022
 #1
avatar+9676 
0

Whenever you have (x+a)n and the exponent is bigger than 2, you want to use binomial theorem to expand it.

Otherwise, you can use sum of square identity (a + b)^2 = a^2 + 2ab + b^2.

 

(x2)2(x+2)3=(x24x+4)((30)x3+(31)x22+(32)x22+(33)23)=(x24x+4)(x3+6x2+12x+8)

 

Afterwards, you just expand it out like this:

 

 

(x24x+4)(x3+6x2+12x+8)=x2(x3+6x2+12x+8)4x(x3+6x2+12x+8)+4(x3+6x2+12x+8)

 

And then expand each clump. It is troublesome, but it will work out nicely.

 May 8, 2022
 #2
avatar+2669 
0

We can write this as: (x2)×(x+2)×(x2)×(x+2)×(x+2)

 

Recall the identity: (ab)(a+b)=a2b2

 

This means we can rewrite the equation as: (x24)×(x24)×(x2)

 

We know that (x24)(x24)=x2×x24×x24×x2+16=x48x2+16

 

Now, we have: (x48x2+16)(x+2)

 

Can you expand this?

 May 9, 2022
 #3
avatar+9676 
0

This method is nicer and is easier to do! 

MaxWong  May 9, 2022

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