7.4 Factor special products  (Page 2/7)

Solution

The first and last terms are squares. See if the middle term fits the pattern of a perfect square trinomial. The middle term is negative, so the binomial square would be ${(a-b)}^2$ .

Are the first and last terms perfect squares?
Check the middle term.
Does is match ${(a-b)}^2$ ? Yes.
Write the square of a binomial.
Check by mulitplying.
${(9y-4)}^2$
${\left(9y\right)}^2-2\cdot 9y\cdot 4+4^2$
$81y^2-72y+16✓$

Solution

Test each term to verify the pattern.
Factor.
Check by mulitplying.
${(6x+7y)}^2$
${\left(6x\right)}^2+2\cdot 6x\cdot 7y+{\left(7y\right)}^2$
$36x^2+84xy+49y^2✓$

Solution

$\begin{array}{cccc}& & & \hfill 9x^2+50x+25\hfill \\ \text{Are the first and last terms perfect squares?}\hfill & & & \hfill {\left(3x\right)}^2{\left(5\right)}^2\hfill \\ \text{Check the middle term—is it}2ab?\hfill & & & \hfill {\left(3x\right)}^2{}_{\text{↘}}\underset{\underset{30x}{2\left(3x\right)\left(5\right)}}{}{}_{\text{↙}}{\left(5\right)}^2\hfill \\ \text{No!}30x\ne 50x\hfill & & & \text{This does not fit the pattern!}\hfill \\ \text{Factor using the “ac” method.}\hfill & & & \hfill 9x^2+50x+25\hfill \\ \\ \\ \\ \text{Notice:}\begin{array}{c}\hfill ac\hfill \\ \hfill 9·25\hfill \\ \hfill 225\hfill \end{array}\text{and}\begin{array}{c}\hfill 5·45=225\hfill \\ \hfill 5+45=50\hfill \end{array}\hfill \\ \text{Split the middle term.}\hfill & & & \hfill 9x^2+5x+45x+25\hfill \\ \text{Factor by grouping.}\hfill & & & \hfill x\left(9x+5\right)+5\left(9x+5\right)\hfill \\ & & & \hfill \left(9x+5\right)\left(x+5\right)\hfill \\ \text{Check.}\hfill & & & \\ \\ \left(9x+5\right)\left(x+5\right)\hfill & & & \\ 9x^2+45x+5x+25\hfill & & & \\ 9x^2+50x+25✓\hfill & & & \end{array}$

Solution

	$36x^{2}y-48xy+16y$
Is there a GCF? Yes, 4 y , so factor it out.	$4y(9x^2-12x+4)$
Is this a perfect square trinomial?
Verify the pattern.
Factor.	$4y{(3x-2)}^2$
Remember: Keep the factor 4 y in the final product.
Check.
$4y{(3x-2)}^2$
$4y[{\left(3x\right)}^2-2·3x·2+2^2]$
$4y{\left(9x\right)}^2-12x+4$
$36x^{2}y-48xy+16y✓$

Solution