More on Factoring General Polynomials

Example 7.38

How to Factor Trinomials Using the "ac" Method

Factor: \(6 x^{2}+7 x+2\).

Solution

Step 1. Factor any GCF.	Is there a greatest common factor? No.	\(6 x^{2}+7 x+2\)
Step 2. Find the product \(a c\).	\(\begin{array}{c} a \cdot c \\ 6 \cdot 2 \\ 12 \end{array} \)	\(ax^2 + bx+c\) \(6x^2+7x+2\)
Step 3. Find two numbers \(m\) and \(n\) that: Multiply to ac \(m \cdot n=a \cdot c\) Add to \(b \quad m+n=b\)	Find two numbers that multiply to 12 and add to 7. Both factors must be positive. \(3 \cdot 4=12 \qquad 3+4=7\)
Step 4. Split the middle term using \(m\), and \(n\) \(a x^{2}+b x+c\) \(\underbrace{bx}_{a x^{2}+m x+n x+c}\)	Rewrite \(7 x\) as \(3 x+4 x\) Notice that \(6 x^{2}+3 x+4 x+2\) is equal to \(6 x^{2}+7 x+2 .\) We just split the middle term to get a more useful form.
Step 5. Factor by grouping.		\( \begin{array}{c} 3 x(2 x+1)+2(2 x+1) \\ (2 x+1)(3 x+2) \end{array} \)
Step 5. Check by multiplying.		\( \begin{array}{l} (2 x+1)(3 x+2) \\ 6 x^{2}+4 x+3 x+2 \\ 6 x^{2}+7 x+2 \text{✓} \end{array} \)

Try It 7.75

Factor: \(6 x^{2}+13 x+2\).

Try It 7.76

Factor: \(4 y^{2}+8 y+3\).

HOW TO

Factor trinomials of the form using the "ac" method.

Step 1. Factor any GCF.

Step 2. Find the product ac.

Step 3. Find two numbers \(m\) and \(n\) that:

\( \begin{array}{l} \text { Multiply to } a c \qquad m \cdot n=a \cdot c\\ \text { Add to } b \qquad \qquad \, m+n=b \end{array} \)

Step 4. Split the middle term using \(m\) and \(n\):\( \begin{array}{c}a x^{2}+b x+c \\ a x^{2}+\overbrace{m x+n x}^{bx}+c \end{array} \)

Step 5. Factor by grouping.

Step 6. Check by multiplying the factors.

Example 7.39

Factor: \(8 u^{2}-17 u-21\).

Solution

Is there a greatest common factor? No.		\(ax^2+bx+c\) \(8u^2-17u-21\)
Find \(a \cdot c\).	\(a \cdot c\)
	\(8(−21)\)
	\(−168\)

Find two numbers that multiply to −168 and add to −17. The larger factor must be negative.

Factors of \(−168\)	Sum of factors
\(1,−168\)	\(1+(−168)=−167\)
\(2,−84\)	\(2+(−84)=−82\)
\(3,−56\)	\(3+(−56)=−53\)
\(4,−42\)	\(4+(−42)=−38\)
\(6,−28\)	\(6+(−28)=−22\)
\(7,−24\)	\( 7+(−24)=−17 \ast \)
\(8,−21\)	\(8+(−21)=−13\)

 

Split the middle term using 7u and -24 u.	\( 8u^2 -17u-21\\ \qquad \quad \downarrow \\ \underbrace{8u^2+7u}_{ } \; \underbrace{-24u-21}_{ } \)
Factor by grouping.	\( \begin{array}{c} u(8 u+7)-3(8 u+7) \\ (8 u+7)(u-3) \end{array} \)
Check by multiplying. \( \begin{array}{l} (8 u+7)(u-3) \\ 8 u^{2}-24 u+7 u-21 \\ 8 u^{2}-17 u-21 \text{✓} \end{array} \)

Try It 7.77

Factor: \(20 h^{2}+13 h-15\).

Try It 7.78

Factor: \(6 g^{2}+19 g-20\).

Example 7.40

Factor: \(2 x^{2}+6 x+5\).

Solution

Is there a greatest common factor? No.	\(ax^2+bx+c\) \(2x^2+6x+5\)
Find \(a \cdot c\).	\(a \cdot c\)
	\(2(5)\)
	\(10\)

 

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

Factors of 10	Sum of factors
\(1,10\)	\(1+10=11\)
\(2, 5\)	\(2+5=7\)

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

Try It 7.79

Factor:\(10 t^{2}+19 t-15\).

Try It 7.80

Factor: \(3 u^{2}+8 u+5\).

Example 7.41

Factor: \(10 y^{2}-55 y+70\).

Solution

Is there a greatest common factor? Yes. The GCF is 5.	\(10 y^{2}-55 y+70\)
Factor it. Be careful to keep the factor of 5 all the way through the solution!	\(5\left(2 y^{2}-11 y+14\right)\)
The trinomial inside the parentheses has a leading coefficient that is not 1.	\(ax^3 + bx + c\) \(5(2y^2 - 11y +14\)
Factor the trinomial.	\(5(y-2)(2 y-7)\)
Check by multiplying all three factors.
\(5\left(2 y^{2}-2 y-4 y+14\right)\)
\(5\left(2 y^{2}-11 y+14\right)\)
\(10 y^{2}-55 y+70 \text{✓}\)

Try It 7.81

Factor: \(16 x^{2}-32 x+12\).

Try It 7.82

Factor: \(18 w^{2}-39 w+18\).

HOW TO

Choose a strategy to factor polynomials completely (updated).

Step 1. Is there a greatest common factor?

• Factor it.

Step 2. Is the polynomial a binomial, trinomial, or are there more than three terms?

• If it is a binomial, right now we have no method to factor it.
• If it is a trinomial of the form \(x^{2}+b x+c\)
  Undo FOIL \((x \quad)(x \quad)\).
• If it is a trinomial of the form \(a x^{2}+b x+c\)
  Use Trial and Error or the "ac" method.
• If it has more than three terms
  Use the grouping method.

Step 3. Check by multiplying the factors.