7.5 General strategy for factoring polynomials

Solution

$\begin{array}{ccccccc}\text{Is there a GCF?}\hfill & & & \text{Yes,}4x^4.\hfill & & & \hfill 4x^5+12x^4\hfill \\ & & & \text{Factor out the GCF.}\hfill & & & \hfill 4x^4\left(x+3\right)\hfill \\ \text{In the parentheses, is it a binomial, a}\hfill & & & & & & \\ \text{trinomial, or are there more than three terms?}\hfill & & & \text{Binomial.}\hfill & & & \\ \text{Is it a sum?}\hfill & & & & & & \text{Yes.}\hfill \\ \text{Of squares? Of cubes?}\hfill & & & & & & \text{No.}\hfill \\ \text{Check.}\hfill & & & & & & \\ \\ \text{Is the expression factored completely?}\hfill & & & & & & \text{Yes.}\hfill \\ \text{Multiply.}\hfill & & & & & & \\ \\ \\ 4x^4\left(x+3\right)\hfill & & & & & & \\ 4x^4·x+4x^4·3\hfill & & & & & & \\ 4x^5+12x^4✓\hfill & & & & & & \end{array}$

Solution

Is there a GCF?	No.
Is it a binomial, trinomial, or are there more than three terms?	Trinomial.
Are a and c perfect squares?	No, a = 12, not a perfect square.
Use trial and error or the “ac” method. We will use trial and error here.

Check.

$\left(3x-2\right)\left(4x-1\right)$

$12x^2-3x-8x+2$

$12x^2-11x+2✓$

Solution

$\begin{array}{ccccccc}\text{Is there a GCF?}\hfill & & & \text{Yes,}g.\hfill & & & \hfill g^3+25g\hfill \\ \text{Factor out the GCF.}\hfill & & & & & & \hfill g\left(g^2+25\right)\hfill \\ \text{In the parentheses, is it a binomial, trinomial,}\hfill & & & & & & \\ \text{or are there more than three terms?}\hfill & & & \text{Binomial.}\hfill & & & \\ \text{Is it a sum ? Of squares?}\hfill & & & \text{Yes.}\hfill & & & \text{Sums of squares are prime.}\hfill \\ \text{Check.}\hfill & & & & & & \\ \\ \text{Is the expression factored completely?}\hfill & & & \text{Yes.}\hfill & & & \\ \text{Multiply.}\hfill & & & & & & \\ g\left(g^2+25\right)\hfill & & & & & & \\ g^3+25g✓\hfill & & & & & & \end{array}$

Solution

$\begin{array}{ccccccc}\text{Is there a GCF?}\hfill & & & \text{Yes, 3.}\hfill & & & \hfill 12y^2-75\hfill \\ \text{Factor out the GCF.}\hfill & & & & & & \hfill 3\left(4y^2-25\right)\hfill \\ \text{In the parentheses, is it a binomial, trinomial},\hfill & & & & & & \\ \text{or are there more than three terms?}\hfill & & & \text{Binomial.}\hfill & & \\ \text{Is it a sum?}\hfill & & & \text{No.}\hfill & & & \\ \text{Is it a difference? Of squares or cubes?}\hfill & & & \text{Yes, squares.}\hfill & & & \hfill 3\left({\left(2y\right)}^2-{\left(5\right)}^2\right)\hfill \\ \text{Write as a product of conjugates.}\hfill & & & & & & \hfill 3\left(2y-5\right)\left(2y+5\right)\hfill \\ \text{Check.}\hfill & & & & & & \\ \\ \text{Is the expression factored completely?}\hfill & & & \text{Yes.}\hfill & & & \\ \text{Neither binomial is a difference of}\hfill & & & & & & \\ \text{squares.}\hfill & & & & & & \\ \text{Multiply.}\hfill & & & & & & \\ 3\left(2y-5\right)\left(2y+5\right)\hfill & & & & & & \\ 3\left(4y^2-25\right)\hfill & & & & & & \\ 12y^2-75✓\hfill & & & & & & \end{array}$