Question

For each expression listed, is it a difference of squares? Explain yes or no and why or why not. Explain in enough detail so there is no doubt that it is or isn't a difference of squares. 1. y5 − 25 2. 9m2n2 − 121 3. p18 − q10 4. 16r2 − 27

Answer 1

Explanation:

If we can write a polynomial in the form $a^2-b^2$, then we can factor it as:

$a^2-b^2=(a-b)(a+b)$

In this problem, we have the following expressions:

$1. \ y^5 - 25 \\ \\ 2. \ 9m^2n^2-121 \\ \\ 3. p^{18} - q^{10} \\ \\ 4. 16r^2 - 27$

• For 1 and 4 we can't write this in the form $a^2-b^2$ so this is not a difference of squares.

• For 2:

$9m^2n^2-121 =3^2m^2n^2-11^2 \\ \\ =(3mn)^2-11^2 =(3mn-11)(3mn+11)$

So we could write 2 as a difference of squares.

• For 3:

$p^{18} - q^{10}=(p^9)^2-(q^5)^2=(p^9-q^5)(p^9+q^5) \\ \\ \\ Remember: \ (a^m)^n=a^{mn}$

So we could write 2 as a difference of squares.