Question

match each polynomial from the first column with a factoring technique in the second column. if none of the techniques can be used to factor the polynomial, choose none.
$49x^2 + 100y^2$
$64a^2 - 48ab + 9b^2$
$a^3 - 3a^2 + 2a - 6$
$144m^2 - 25n^2$
factor out gcf perfect square trinomial difference of squares factor by grouping none

Explanation:

Step1: Analyze \(49x^{2}+100y^{2}\)

The expression is a sum of two squares (\((7x)^{2}+(10y)^{2}\)). The difference of squares can be factored, but the sum of squares (in real numbers) cannot be factored with the given techniques. So it matches "None".

Step2: Analyze \(64a^{2}-48ab + 9b^{2}\)

Check if it's a perfect square trinomial. A perfect square trinomial has the form \(A^{2}-2AB + B^{2}=(A - B)^{2}\). Here, \(A = 8a\), \(B=3b\), and \(2AB=2\times8a\times3b = 48ab\). So \(64a^{2}-48ab + 9b^{2}=(8a - 3b)^{2}\), which matches "Perfect Square Trinomial".

Step3: Analyze \(a^{3}-3a^{2}+2a - 6\)

We use factoring by grouping. Group the first two terms and the last two terms: \((a^{3}-3a^{2})+(2a - 6)\). Factor out \(a^{2}\) from the first group and 2 from the second group: \(a^{2}(a - 3)+2(a - 3)\). Then factor out \((a - 3)\) to get \((a - 3)(a^{2}+2)\). So it matches "Factor by Grouping".

Step4: Analyze \(144m^{2}-25n^{2}\)

This is a difference of squares, since \(144m^{2}=(12m)^{2}\) and \(25n^{2}=(5n)^{2}\). The difference of squares formula is \(A^{2}-B^{2}=(A + B)(A - B)\), so \(144m^{2}-25n^{2}=(12m + 5n)(12m - 5n)\), which matches "Difference of Squares".

Answer:

• \(49x^{2}+100y^{2}\): None
• \(64a^{2}-48ab + 9b^{2}\): Perfect Square Trinomial
• \(a^{3}-3a^{2}+2a - 6\): Factor by Grouping
• \(144m^{2}-25n^{2}\): Difference of Squares