Question

Factor the expression 36a^4 b^10 - 8 1a^16 b^20 using the two different techniques listed for Parts 1(a) and 1 (b).(a) Factor the given expression using the GCF monomial.(b) Factor the given expression using the difference of squares.

Answer 1

In this problem, we need to factor a given binomial using two different techniques.

We are given

$36a^4b^{10}-81a^{16}b^{20}$

GCF Technique

The first technique we'll use is finding the greatest common factor of both terms. We'll look at each constant and variable separately.

$\begin{gathered} GCF(36,81)=9 \\ \\ GCF(a^4,a^{16})=a^4 \\ \\ GCF(b^{10},b^{20})=b^{10} \end{gathered}$

Therefore, the overall GCF of the binomial is:

$9a^4b^{10}$

We can factor it from each term to get:

$9a^4b^{10}(4-9a^{12}b^{10})$

Difference of Squares

In the second technique, we are going to apply the difference of squares.

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

To get it into that form, we need to rewrite each term of the binomial as a square:

$\begin{gathered} 36a^4b^{10}\rightarrow(6a^2b^5)^2 \\ \\ 81a^{16}b^{20}\rightarrow(9a^8b^{10})^2 \end{gathered}$

Now we can write it as

$(6a^2b^5)^2-(9a^8b^{10})^2$

Using our rule, we get

$(6a^2b^5+9a^8b^{10})(6a^2b^5-9a^8b^{10})$