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

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

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.