7.3 Factor quadratic trinomials with leading coefficient other than  (Page 4/6)

Solution

	$10y^4+55y^3+60y^2$
Notice the greatest common factor, and factor it first.	$15y^2(2y^2+11y+12)$
Factor the trinomial.

Consider all the combinations.

$\begin{array}{cccc}\text{The correct factors are those whose product}\hfill & & & \hfill 5y^2\left(y+4\right)\left(2y+3\right)\hfill \\ \text{is the original trinomial. Remember to include}\hfill & & & \\ \text{the factor}5y^2.\hfill & & & \\ \text{Check by multiplying.}\hfill & & & \\ \\ \\ 5y^2\left(y+4\right)\left(2y+3\right)\hfill & & & \\ 5y^2\left(2y^2+8y+3y+12\right)\hfill & & & \\ 10y^4+55y^3+60y^2✓\hfill & & & \end{array}$

Solution

Solution

Is there a greatest common factor? No.
Find $a\cdot c$ .	$a\cdot c$
	$8\left(\mathrm{-21}\right)$
	$\mathrm{-168}$

Find two numbers that multiply to $\mathrm{-168}$ and add to $\mathrm{-17}$ . The larger factor must be negative.

$\begin{array}{cccc}\text{Split the middle term using}7u\text{and}\mathrm{-24}u.\hfill & & & \hfill 8u^2-\underset{\text{↙}}{17}\underset{\text{↘}}{u}-21\hfill \\ & & & \hfill \underbracket{8u^2+7u}\underbracket{\mathrm{-24}u-21}\hfill \\ \\ \\ \text{Factor by grouping.}\hfill & & & \hfill u\left(8u+7\right)-3\left(8u+7\right)\hfill \\ & & & \hfill \left(8u+7\right)\left(u-3\right)\hfill \\ \text{Check by multiplying.}\hfill & & & \\ \\ \\ \left(8u+7\right)\left(u-3\right)\hfill & & & \\ 8u^2-24u+7u-21\hfill & & & \\ 8u^2-17u-21✓\hfill & & & \end{array}$

Solution

Is there a greatest common factor? No.
Find $a\cdot c$ .	$a\cdot c$
	$2\left(5\right)$
	$10$

Find two numbers that multiply to 10 and add to 6.

There are no factors that multiply to 10 and add to 6. The polynomial is prime.

Solution

Is there a greatest common factor? Yes. The GCF is 5.
Factor it. Be careful to keep the factor of 5 all the way through the solution!
The trinomial inside the parentheses has a leading coefficient that is not 1.
Factor the trinomial.
Check by mulitplying all three factors.
$5(2y^2-2y-4y+14)$
$5(2y^2-11y+14)$
$10y^2-55y+70✓$