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?
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.
(x−2)2(x+2)3=(x2−4x+4)((30)x3+(31)x2⋅2+(32)x⋅22+(33)23)=(x2−4x+4)(x3+6x2+12x+8)
Afterwards, you just expand it out like this:
(x2−4x+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.
We can write this as: (x−2)×(x+2)×(x−2)×(x+2)×(x+2)
Recall the identity: (a−b)(a+b)=a2−b2
This means we can rewrite the equation as: (x2−4)×(x2−4)×(x−2)
We know that (x2−4)(x2−4)=x2×x2−4×x2−4×x2+16=x4−8x2+16
Now, we have: (x4−8x2+16)(x+2).
Can you expand this?