6.4 General Strategy for Factoring Polynomials

$\begin{array}{cccccc}& & & & & 7x^3-21x^2-70x\hfill \\ \text{Is there a GCF? Yes,}7x.\hfill & & & & & \\ \text{Factor out the GCF.}\hfill & & & & & 7x\left(x^2-3x-10\right)\hfill \\ \begin{array}{c}\text{In the parentheses, is it a binomial, trinomial,}\hfill \\ \text{or are there more terms?}\hfill \\ \text{Trinomial with leading coefficient 1.}\hfill \end{array}\hfill & & & & & \\ \text{“Undo” FOIL.}\hfill & & & & & 7x\left(x\right)\left(x\right)\hfill \\ & & & & & 7x\left(x+2\right)\left(x-5\right)\hfill \\ \text{Is the expression factored completely? Yes.}\hfill & & & & & \\ \text{Neither binomial can be factored.}\hfill & & & & & \\ \text{Check your answer.}\hfill & & & & & \\ \text{Multiply.}\hfill & & & & & \\ \\ \\ \hfill 7x\left(x+2\right)\left(x-5\right)\hfill & & & & & \\ \\ \hfill 7x\left(x^2-5x+2x-10\right)\hfill & & & & & \\ \hfill 7x\left(x^2-3x-10\right)\hfill & & & & & \\ \hfill 7x^3-21x^2-70x✓\hfill & & & & & \end{array}$

$\begin{array}{cccccc}& & & & & \hfill 24y^2-150\hfill \\ \text{Is there a GCF? Yes, 6.}\hfill & & & & & \\ \text{Factor out the GCF.}\hfill & & & & & \hfill 6\left(4y^2-25\right)\hfill \\ \text{In the parentheses, is it a binomial, trinomial}\hfill & & & & & \\ \text{or are there more than three terms? Binomial.}\hfill & & & & & \\ \text{Is it a sum? No.}\hfill & & & & & \\ \text{Is it a difference? Of squares or cubes? Yes, squares.}\hfill & & & & & \hfill 6\left({\left(2y\right)}^2-{\left(5\right)}^2\right)\hfill \\ \text{Write as a product of conjugates.}\hfill & & & & & \hfill 6\left(2y-5\right)\left(2y+5\right)\hfill \\ \\ \\ \text{Is the expression factored completely?}\hfill & & & & & \\ \text{Neither binomial can be factored.}\hfill & & & & & \\ \text{Check:}\hfill & & & & & \\ \\ \\ \text{Multiply.}\hfill & & & & & \\ \\ \hfill 6\left(2y-5\right)\left(2y+5\right)\hfill & & & & & \\ \\ \hfill 6\left(4y^2-25\right)\hfill & & & & & \\ \hfill 24y^2-150✓\hfill & & & & & \end{array}$

$\begin{array}{cccccc}& & & & & \hfill 4a^2-12ab+9b^2\hfill \\ \text{Is there a GCF? No.}\hfill & & & & & \\ \text{Is it a binomial, trinomial, or are there more terms?}\hfill & & & & & \\ \text{Trinomial with}a\ne 1.\text{But the first term is a perfect square.}\hfill & & & & & \\ \text{Is the last term a perfect square? Yes.}\hfill & & & & & \hfill {\left(2a\right)}^2-12ab+{\left(3b\right)}^2\hfill \\ \text{Does it fit the pattern,}{a}^2-2ab+b^2?\text{Yes.}\hfill & & & & & \hfill {\left(2a\right)}^2{}_{\text{↘}}\underset{\mathrm{-2}\left(2a\right)\left(3b\right)}{\mathrm{-12}ab+}{}_{\text{↙}}{\left(3b\right)}^2\hfill \\ \text{Write it as a square.}\hfill & & & & & \hfill {\left(2a-3b\right)}^2\hfill \\ \\ \\ \text{Is the expression factored completely? Yes.}\hfill & & & & & \\ \text{The binomial cannot be factored.}\hfill & & & & & \\ \text{Check your answer.}\hfill & & & & & \\ \\ \\ \text{Multiply.}\hfill & & & & & \\ \hfill {\left(2a-3b\right)}^2\hfill & & & & & \\ \hfill {\left(2a\right)}^2-2·2a·3b+{\left(3b\right)}^2\hfill & & & & & \\ \hfill 4a^2-12ab+9b^2✓\hfill & & & & & \end{array}$

$\begin{array}{cccccc}& & & & & \hfill 12x^3{y}^2+75x{y}^2\hfill \\ \text{Is there a GCF? Yes,}3x{y}^2.\hfill & & & & & \\ \text{Factor out the GCF.}\hfill & & & & & \hfill 3x{y}^2\left(4x^2+25\right)\hfill \\ \text{In the parentheses, is it a binomial, trinomial, or are}\hfill & & & & & \\ \text{there more than three terms? Binomial.}\hfill & & & & & \\ \\ \text{Is it a sum? Of squares? Yes.}\hfill & & & & & \text{Sums of squares are prime.}\hfill \\ \\ \\ \text{Is the expression factored completely? Yes.}\hfill & & & & & \\ \text{Check:}\hfill & & & & & \\ \\ \\ \text{Multiply.}\hfill & & & & & \\ \hfill 3x{y}^2\left(4x^2+25\right)\hfill & & & & & \\ \hfill 12x^3{y}^2+75x{y}^2✓\hfill & & & & & \end{array}$

$\begin{array}{cccccc}& & & & & \hfill 3x^{5}y-48xy\hfill \\ \text{Is there a GCF? Factor out}3xy\hfill & & & & & \hfill 3xy\left(x^4-16\right)\hfill \\ \begin{array}{c}\text{Is the binomial a sum or difference? Of squares or cubes?}\hfill \\ \text{Write it as a difference of squares.}\hfill \end{array}\hfill & & & & & \hfill 3xy\left({\left(x^2\right)}^2-{\left(4\right)}^2\right)\hfill \\ \text{Factor it as a product of conjugates}\hfill & & & & & \hfill 3xy\left(x^2-4\right)\left(x^2+4\right)\hfill \\ \text{The first binomial is again a difference of squares.}\hfill & & & & & \hfill 3xy\left({\left(x\right)}^2-{\left(2\right)}^2\right)\left(x^2+4\right)\hfill \\ \text{Factor it as a product of conjugates.}\hfill & & & & & \hfill 3xy\left(x-2\right)\left(x+2\right)\left(x^2+4\right)\hfill \\ \text{Is the expression factored completely? Yes.}\hfill & & & & & \\ \text{Check your answer.}\hfill & & & & & \\ \text{Multiply.}\hfill & & & & & \\ \\ \\ \hfill 3xy\left(x-2\right)\left(x+2\right)\left(x^2+4\right)\hfill & & & & & \\ \hfill 3xy\left(x^2-4\right)\left(x^2+4\right)\hfill & & & & & \\ \hfill 3xy\left(x^4-16\right)\hfill & & & & & \\ \hfill 3x^{5}y-48xy✓\hfill & & & & & \end{array}$

$\begin{array}{cccccc}& & & & & \hfill 4x^2+8bx-4ax-8ab\hfill \\ \text{Is there a GCF? Factor out the GCF, 4.}\hfill & & & & & \hfill 4\left(x^2+2bx-ax-2ab\right)\hfill \\ \text{There are four terms. Use grouping.}\hfill & & & & & \hfill 4\left[x\left(x+2b\right)-a\left(x+2b\right)\right]\hfill \\ & & & & & \hfill 4\left(x+2b\right)\left(x-a\right)\hfill \\ \text{Is the expression factored completely? Yes.}\hfill & & & & & \\ \text{Check your answer.}\hfill & & & & & \\ \text{Multiply.}\hfill & & & & & \\ \\ \\ \hfill 4\left(x+2b\right)\left(x-a\right)\hfill & & & & & \\ \hfill 4\left(x^2-ax+2bx-2ab\right)\hfill & & & & & \\ \hfill 4x^2+8bx-4ax-8ab✓\hfill & & & & & \end{array}$

$\begin{array}{cccccc}& & & & & \hfill 40x^{2}y+44xy-24y\hfill \\ \text{Is there a GCF? Factor out the GCF,}4y.\hfill & & & & & \hfill 4y\left(10x^2+11x-6\right)\hfill \\ \text{Factor the trinomial with}a\ne 1.\hfill & & & & & \hfill 4y\left(10x^2+11x-6\right)\hfill \\ & & & & & \hfill 4y\left(5x-2\right)\left(2x+3\right)\hfill \\ \text{Is the expression factored completely? Yes.}\hfill & & & & & \\ \text{Check your answer.}\hfill & & & & & \\ \text{Multiply.}\hfill & & & & & \\ \\ \\ \hfill 4y\left(5x-2\right)\left(2x+3\right)\hfill & & & & & \\ \hfill 4y\left(10x^2+11x-6\right)\hfill & & & & & \\ \hfill 40x^{2}y+44xy-24y✓\hfill & & & & & \end{array}$

$\begin{array}{cccccc}& & & & & \hfill 9x^2-12xy+4y^2-49\hfill \\ \text{Is there a GCF? No.}\hfill & & & & & \\ \begin{array}{c}\text{With more than 3 terms, use grouping. Last 2 terms}\hfill \\ \text{have no GCF. Try grouping first 3 terms.}\hfill \end{array}\hfill & & & & & \hfill 9x^2-12xy+4y^2-49\hfill \\ \begin{array}{c}\text{Factor the trinomial with}a\ne 1.\text{But the first term is a}\hfill \\ \text{perfect square.}\hfill \end{array}\hfill & & & & & \\ \text{Is the last term of the trinomial a perfect square? Yes.}\hfill & & & & & \hfill {\left(3x\right)}^2-12xy+{\left(2y\right)}^2-49\hfill \\ \text{Does the trinomial fit the pattern,}{a}^2-2ab+b^2?\text{Yes.}\hfill & & & & & \hfill {\left(3x\right)}^2{}_{\text{↘}}\underset{\mathrm{-2}\left(3x\right)\left(2y\right)}{\mathrm{-12}xy+}{}_{\text{↙}}{\left(2y\right)}^2-49\hfill \\ \text{Write the trinomial as a square.}\hfill & & & & & \hfill {\left(3x-2y\right)}^2-49\hfill \\ \begin{array}{c}\text{Is this binomial a sum or difference? Of squares or}\hfill \\ \text{cubes? Write it as a difference of squares.}\hfill \end{array}\hfill & & & & & \hfill {\left(3x-2y\right)}^2-7^2\hfill \\ \text{Write it as a product of conjugates.}\hfill & & & & & \hfill \left(\left(3x-2y\right)-7\right)\left(\left(3x-2y\right)+7\right)\hfill \\ & & & & & \hfill \left(3x-2y-7\right)\left(3x-2y+7\right)\hfill \\ \text{Is the expression factored completely? Yes.}\hfill & & & & & \\ \text{Check your answer.}\hfill & & & & & \\ \text{Multiply.}\hfill & & & & & \\ \\ \\ \hfill \left(3x-2y-7\right)\left(3x-2y+7\right)\hfill & & & & & \\ \hfill 9x^2-6xy-21x-6xy+4y^2+14y+21x-14y-49\hfill & & & & & \\ \hfill 9x^2-12xy+4y^2-49✓\hfill & & & & & \end{array}$