### Key Concepts

### 6.1 Greatest Common Factor and Factor by Grouping

**How to find the greatest common factor (GCF) of two expressions.**- Step 1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
- Step 2. List all factors—matching common factors in a column. In each column, circle the common factors.
- Step 3. Bring down the common factors that all expressions share.
- Step 4. Multiply the factors.

**Distributive Property:**If*a*,*b*, and*c*are real numbers, then

$$a\left(b+c\right)=ab+ac\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}ab+ac=a\left(b+c\right)$$

The form on the left is used to multiply. The form on the right is used to factor.**How to factor the greatest common factor from a polynomial.**- Step 1. Find the GCF of all the terms of the polynomial.
- Step 2. Rewrite each term as a product using the GCF.
- Step 3. Use the “reverse” Distributive Property to factor the expression.
- Step 4. Check by multiplying the factors.

**Factor as a Noun and a Verb:**We use “factor” as both a noun and a verb.

$$\begin{array}{cccc}\text{Noun:}\hfill & & & \text{7 is a}\phantom{\rule{0.2em}{0ex}}{\text{factor}}\phantom{\rule{0.2em}{0ex}}\text{of 14}\hfill \\ \text{Verb:}\hfill & & & {\text{factor}}\phantom{\rule{0.2em}{0ex}}\text{3 from}\phantom{\rule{0.2em}{0ex}}3a+3\hfill \end{array}$$**How to factor by grouping.**- Step 1. Group terms with common factors.
- Step 2. Factor out the common factor in each group.
- Step 3. Factor the common factor from the expression.
- Step 4. Check by multiplying the factors.

### 6.2 Factor Trinomials

**How to factor trinomials of the form ${x}^{2}+bx+c.$**- Step 1. Write the factors as two binomials with first terms
*x*. $\phantom{\rule{4em}{0ex}}\begin{array}{c}\hfill {x}^{2}+bx+c\hfill \\ \hfill \left(x\phantom{\rule{1.2em}{0ex}}\right)\left(x\phantom{\rule{1.2em}{0ex}}\right)\hfill \end{array}$ - Step 2. Find two numbers
*m*and*n*that

$\begin{array}{ccc}\text{multiply to}\hfill & & c,m\xb7n=c\hfill \\ \text{add to}\hfill & & b,m+n=b\hfill \end{array}$ - Step 3. Use
*m*and*n*as the last terms of the factors. $\phantom{\rule{8em}{0ex}}\left(x+m\right)\left(x+n\right)$ - Step 4. Check by multiplying the factors.

- Step 1. Write the factors as two binomials with first terms
**Strategy for Factoring Trinomials of the Form ${x}^{2}+bx+c$:**When we factor a trinomial, we look at the signs of its terms first to determine the signs of the binomial factors.

$\begin{array}{c}\hfill \begin{array}{c}\hfill {x}^{2}+bx+c\hfill \\ \hfill \left(x+m\right)\left(x+n\right)\hfill \end{array}\hfill \\ \hfill \mathbf{\text{When}}\phantom{\rule{0.2em}{0ex}}{\text{c}}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{is positive,}}\phantom{\rule{0.2em}{0ex}}{\text{m}}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{and}}\phantom{\rule{0.2em}{0ex}}{\text{n}}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{have the same sign.}}\hfill \\ \hfill b\phantom{\rule{0.2em}{0ex}}\text{positive}\phantom{\rule{16.5em}{0ex}}b\phantom{\rule{0.2em}{0ex}}\text{negative}\hfill \\ \hfill m,n\phantom{\rule{0.2em}{0ex}}\text{positive}\phantom{\rule{15em}{0ex}}m,n\phantom{\rule{0.2em}{0ex}}\text{negative}\hfill \\ \hfill {x}^{2}+5x+6\phantom{\rule{16em}{0ex}}{x}^{2}-6x+8\hfill \\ \hfill \left(x+2\right)\left(x+3\right)\phantom{\rule{15em}{0ex}}\left(x-4\right)\left(x-2\right)\hfill \\ \hfill \text{same signs}\phantom{\rule{16em}{0ex}}\text{same signs}\hfill \\ \hfill \mathbf{\text{When}}\phantom{\rule{0.2em}{0ex}}{\text{c}}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{is negative,}}\phantom{\rule{0.2em}{0ex}}{\text{m}}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{and}}\phantom{\rule{0.2em}{0ex}}{\text{n}}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{have opposite signs.}}\hfill \\ \hfill {x}^{2}+x-12\phantom{\rule{15em}{0ex}}{x}^{2}-2x-15\hfill \\ \hfill \left(x+4\right)\left(x-3\right)\phantom{\rule{15em}{0ex}}\left(x-5\right)\left(x+3\right)\hfill \\ \hfill \text{opposite signs}\phantom{\rule{15em}{0ex}}\text{opposite signs}\hfill \end{array}$

Notice that, in the case when*m*and*n*have opposite signs, the sign of the one with the larger absolute value matches the sign of*b*.**How to factor trinomials of the form $a{x}^{2}+bx+c$ using trial and error.**- Step 1. Write the trinomial in descending order of degrees as needed.
- Step 2. Factor any GCF.
- Step 3. Find all the factor pairs of the first term.
- Step 4. Find all the factor pairs of the third term.
- Step 5. Test all the possible combinations of the factors until the correct product is found.
- Step 6. Check by multiplying.

**How to factor trinomials of the form $a{x}^{2}+bx+c$ using the “ac” method.**- Step 1. Factor any GCF.
- Step 2. Find the product
*ac*. - Step 3. Find two numbers
*m*and*n*that:

$\begin{array}{cccccc}\text{Multiply to}\phantom{\rule{0.2em}{0ex}}ac.\hfill & & & & & m\xb7n=a\xb7c\hfill \\ \text{Add to}\phantom{\rule{0.2em}{0ex}}b.\hfill & & & & & m+n=b\hfill \\ & & & & & a{x}^{2}+bx+c\hfill \end{array}$ - Step 4. Split the middle term using
*m*and*n*. $\phantom{\rule{4em}{0ex}}a{x}^{2}+mx+nx+c$ - Step 5. Factor by grouping.
- Step 6. Check by multiplying the factors.

### 6.3 Factor Special Products

**Perfect Square Trinomials Pattern:**If*a*and*b*are real numbers,

$$\begin{array}{c}\hfill {a}^{2}+2ab+{b}^{2}={\left(a+b\right)}^{2}\hfill \\ \hfill {a}^{2}-2ab+{b}^{2}={\left(a-b\right)}^{2}\hfill \end{array}$$**How to factor perfect square trinomials.**

$\begin{array}{cccccccc}\text{Step 1.}\hfill & \text{Does the trinomial fit the pattern?}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{a}^{2}+2ab+{b}^{2}\hfill & & & \hfill {a}^{2}-2ab+{b}^{2}\hfill \\ & \text{Is the first term a perfect square?}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{\left(a\right)}^{2}\hfill & & & \hfill {\left(a\right)}^{2}\hfill \\ & \text{Write it as a square.}\hfill & & & & & & \\ & \text{Is the last term a perfect square?}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{\left(a\right)}^{2}\phantom{\rule{4.5em}{0ex}}{\left(b\right)}^{2}\hfill & & & \hfill {\left(a\right)}^{2}\phantom{\rule{4.5em}{0ex}}{\left(b\right)}^{2}\hfill \\ & \text{Write it as a square.}\hfill & & & & & & \\ & \text{Check the middle term. Is it}\phantom{\rule{0.2em}{0ex}}2ab?\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{\left(a\right)}^{2}{}_{\text{\u2198}}\underset{2\xb7a\xb7b}{}{}_{\text{\u2199}}{\left(b\right)}^{2}\hfill & & & \hfill {\left(a\right)}^{2}{}_{\text{\u2198}}\underset{2\xb7a\xb7b}{}{}_{\text{\u2199}}{\left(b\right)}^{2}\hfill \\ \text{Step 2.}\hfill & \text{Write the square of the binomial.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{\left(a+b\right)}^{2}\hfill & & & \hfill {\left(a-b\right)}^{2}\hfill \\ \text{Step 3.}\hfill & \text{Check by multiplying.}\hfill & & & & & & \end{array}$**Difference of Squares Pattern:**If $a,b$ are real numbers,

**How to factor differences of squares.**

$\begin{array}{ccccc}\text{Step 1.}\hfill & \text{Does the binomial fit the pattern?}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{a}^{2}-{b}^{2}\hfill \\ & \text{Is this a difference?}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\text{\_\_\_\_}-\text{\_\_\_\_}\hfill \\ & \text{Are the first and last terms perfect squares?}\hfill & & & \\ \text{Step 2.}\hfill & \text{Write them as squares.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{\left(a\right)}^{2}-{\left(b\right)}^{2}\hfill \\ \text{Step 3.}\hfill & \text{Write the product of conjugates.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\left(a-b\right)\left(a+b\right)\hfill \\ \text{Step 4.}\hfill & \text{Check by multiplying.}\hfill & & & \end{array}$**Sum and Difference of Cubes Pattern**

$\begin{array}{c}\hfill {a}^{3}+{b}^{3}=\left(a+b\right)\left({a}^{2}-ab+{b}^{2}\right)\hfill \\ \hfill {a}^{3}-{b}^{3}=\left(a-b\right)\left({a}^{2}+ab+{b}^{2}\right)\hfill \end{array}$**How to factor the sum or difference of cubes.**- Step 1. Does the binomial fit the sum or difference of cubes pattern?

Is it a sum or difference?

Are the first and last terms perfect cubes? - Step 2. Write them as cubes.
- Step 3. Use either the sum or difference of cubes pattern.
- Step 4. Simplify inside the parentheses
- Step 5. Check by multiplying the factors.

- Step 1. Does the binomial fit the sum or difference of cubes pattern?

### 6.4 General Strategy for Factoring Polynomials

**How to use a general strategy for factoring polynomials.**- Step 1. Is there a greatest common factor?

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

If it is a binomial:

Is it a sum?

Of squares? Sums of squares do not factor.

Of cubes? Use the sum of cubes pattern.

Is it a difference?

Of squares? Factor as the product of conjugates.

Of cubes? Use the difference of cubes pattern.

If it is a trinomial:

Is it of the form ${x}^{2}+bx+c?$ Undo FOIL.

Is it of the form $a{x}^{2}+bx+c?$

If*a*and*c*are squares, check if it fits the trinomial square pattern.

Use the trial and error or “ac” method.

If it has more than three terms:

Use the grouping method. - Step 3. Check.

Is it factored completely?

Do the factors multiply back to the original polynomial?

- Step 1. Is there a greatest common factor?

### 6.5 Polynomial Equations

**Polynomial Equation:**A polynomial equation is an equation that contains a polynomial expression. The degree of the polynomial equation is the degree of the polynomial.**Quadratic Equation:**An equation of the form $a{x}^{2}+bx+c=0$ is called a quadratic equation.

$$a,b,c\phantom{\rule{0.2em}{0ex}}\text{are real numbers and}\phantom{\rule{0.2em}{0ex}}a\ne 0$$**Zero Product Property:**If $a\xb7b=0,$ then either $a=0$ or $b=0$ or both.**How to use the Zero Product Property**- Step 1. Set each factor equal to zero.
- Step 2. Solve the linear equations.
- Step 3. Check.

**How to solve a quadratic equation by factoring.**- Step 1. Write the quadratic equation in standard form, $a{x}^{2}+bx+c=0.$
- Step 2. Factor the quadratic expression.
- Step 3. Use the Zero Product Property.
- Step 4. Solve the linear equations.
- Step 5. Check. Substitute each solution separately into the original equation.

**Zero of a Function:**For any function*f*, if $f\left(x\right)=0,$ then*x*is a zero of the function.**How to use a problem solving strategy to solve word problems.**- Step 1.
**Read**the problem. Make sure all the words and ideas are understood. - Step 2.
**Identify**what we are looking for. - Step 3.
**Name**what we are looking for. Choose a variable to represent that quantity. - Step 4.
**Translate**into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation. - Step 5.
**Solve**the equation using appropriate algebra techniques. - Step 6.
**Check**the answer in the problem and make sure it makes sense. - Step 7.
**Answer**the question with a complete sentence.

- Step 1.