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# Proving Polynomial Identities

## Proving Polynomial Identities 1.9 - Solution

a
The difference of squares has the form $(a + b)(a - b) = a^2 - b^2$ and we see that our expression has the same form, with $a=x^2$ and $b = 12$.
$\left(x^2 - 12\right)\left(x^2 + 12\right)$
$\left(x^2\right)^2 - 12^2$
$\left(x^2\right)^2 - 144$
$x^4 - 144$
b
We'll proceed in the same way as before and rewrite the expression using the difference of squares. Remember that the whole of the term $2y$, including the $2$, must be squared.
$\left(2^3+2y\right)\left(2^3 - 2y\right)$
$\left(8+2y\right)\left(8 - 2y\right)$
$8^2 - \left(2y\right)^2$
$8^2 - 2^2y^2$
$64 - 4y^2$
c
Here, the terms are not placed in the same order in the parentheses. However, if we start with simplifying the first parentheses and then change the order of the terms in $(\text{-} a + 2)$, we can use the difference of squares here as well.
$(2 -(\text{-} a))(\text{-} a + 2)$
$(2+a)(\text{-} a + 2)$
$(2+a)(2-a)$
$2^2 - a^2$
$4 - a^2$