Proving Polynomial Identities

Choose Course

Algebra 2

Polynomial Functions

Proving Polynomial Identities expand_more

close

Proving Polynomial Identities 1.9 - Solution

arrow_back Return to Proving Polynomial Identities

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 $b = 12$.

$\left(x^2 - 12\right)\left(x^2 + 12\right)$

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

$\left(x^2\right)^2 - 12^2$

CalcPowCalculate power

$\left(x^2\right)^2 - 144$

PowPow$\left(a^{m}\right)^{n}=a^{m\cdot n}$

$x^4 - 144$

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.

$\left(2^3+2y\right)\left(2^3 - 2y\right)$

CalcPowCalculate power

$\left(8+2y\right)\left(8 - 2y\right)$

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

$8^2 - \left(2y\right)^2$

PowProdII$\left(a b\right)^{m}=a^m b^m$

$8^2 - 2^2y^2$

CalcPowCalculate power

$64 - 4y^2$

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.

$(2 -(\text{-} a))(\text{-} a + 2)$

SubNeg$a-(\text{-} b)=a+b$

$(2+a)(\text{-} a + 2)$

CommutativePropAddCommutative Property of Addition

$(2+a)(2-a)$

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

$2^2 - a^2$

CalcPowCalculate power

$4 - a^2$