Question

Angelina factored (x - 4)8 and wrote that it was equal to (x2 - 42)2(x2 + 42). Use complete sentences to explain how you could confirm whether Angela’s solution is correct. Prove your conclusion mathematically.

Answer 1

Given expression: $(x - 4)^8$.

Angelina wrote : $(x^2 - 4^2)^2(x^2 + 4^2)$.

Let us multipl$(x^2 - 4^2)^2(x^2 + 4^2)=\left(x^2-16\right)^2\left(x^2+16\right)$

$\mathrm{Apply\:Perfect\:Square\:Formula}:\quad \left(a-b\right)^2=a^2-2ab+b^2$

$=\left(x^2\right)^2-2x^2\cdot \:16+16^2$

$\:\left(x^2\right)^2-2x^2\cdot \:16+16^2:\quad x^4-32x^2+256$

$=(x^4-32x^2+256)(\left x^2+16\right)$

$\mathrm{Distribute\:parentheses}$

$=x^4x^2+x^4\cdot \:16+\left(-32x^2\right)x^2+\left(-32x^2\right)\cdot \:16+256x^2+256\cdot \:16$

$=x^6-16x^4-256x^2+4096$

We can see that first term is x^6.

But for  (x - 4)^8 expression, first term should be x^8.

So, we could say that Angela’s solution is incorrect.

Answer 2

Angela's solution is incorrect b/c when multiplied you get a completely different answer then the one she had.

(x^2 - 4^2)^2 = x^4 - 4^4
(x^4 - 4^4) (x^2 + 4^2) = x^2 - 4^6