### Key Concepts

## 7.1 Greatest Common Factor and Factor by Grouping

**Finding the Greatest Common Factor (GCF):**To find the 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 as in Example 7.2.

**Factor the Greatest Common Factor from a Polynomial:**To factor a 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 as in Example 7.5.

**Factor by Grouping:**To factor a polynomial with 4 four or more terms- 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 as in Example 7.15.

## 7.2 Factor Quadratic Trinomials with Leading Coefficient 1

**Factor trinomials of the form ${x}^{2}+bx+c$**- Step 1.
Write the factors as two binomials with first terms
*x*: $\left(x\phantom{\rule{1em}{0ex}}\right)\left(x\phantom{\rule{1em}{0ex}}\right)$. - Step 2.
Find two numbers
*m*and*n*that

Multiply to*c*, $m\xb7n=c$

Add to*b*, $m+n=b$ - Step 3.
Use
*m*and*n*as the last terms of the factors: $(x+m)(x+n)$. - Step 4. Check by multiplying the factors.

- Step 1.
Write the factors as two binomials with first terms

## 7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1

**Factor Trinomials of the Form $a{x}^{2}+bx+c$ using Trial and Error:**See Example 7.33.- Step 1. Write the trinomial in descending order of degrees.
- Step 2. Find all the factor pairs of the first term.
- Step 3. Find all the factor pairs of the third term.
- Step 4. Test all the possible combinations of the factors until the correct product is found.
- Step 5. Check by multiplying.

**Factor Trinomials of the Form $a{x}^{2}+bx+c$ Using the “ac” Method:**See Example 7.38.- Step 1. Factor any GCF.
- Step 2. Find the product ac.
- Step 3.
Find two numbers
*m*and*n*that:

$\begin{array}{cccc}\text{Multiply to}\phantom{\rule{0.2em}{0ex}}ac\hfill & & & m\xb7n=a\xb7c\hfill \\ \text{Add to}\phantom{\rule{0.2em}{0ex}}b\hfill & & & m+n=b\hfill \end{array}$ - Step 4.
Split the middle term using
*m*and*n*:

- Step 5. Factor by grouping.
- Step 6. Check by multiplying the factors.

**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 ${x}^{2}+bx+c$

Undo FOIL $\left(x\phantom{\rule{1em}{0ex}}\right)\left(x\phantom{\rule{1em}{0ex}}\right)$.

If it is a trinomial of the form $a{x}^{2}+bx+c$

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.

## 7.4 Factor Special Products

**Factor perfect square trinomials**See Example 7.42.

$\begin{array}{ccccccc}\text{Step 1.}\phantom{\rule{0.2em}{0ex}}\text{Does the trinomial fit the pattern?}\hfill & & & \hfill {a}^{2}+2ab+{b}^{2}\hfill & & & \hfill {a}^{2}-2ab+{b}^{2}\hfill \\ \phantom{\rule{3em}{0ex}}\text{Is the first term a perfect square?}\hfill & & & \hfill {\left(a\right)}^{2}\hfill & & & \hfill {\left(a\right)}^{2}\hfill \\ \phantom{\rule{3em}{0ex}}\text{Write it as a square.}\hfill & & & & & & \\ \phantom{\rule{3em}{0ex}}\text{Is the last term a perfect square?}\hfill & & & \hfill {\left(a\right)}^{2}\phantom{\rule{4.5em}{0ex}}{\left(b\right)}^{2}\hfill & & & \hfill {\left(a\right)}^{2}\phantom{\rule{4.5em}{0ex}}{\left(b\right)}^{2}\hfill \\ \phantom{\rule{3em}{0ex}}\text{Write it as a square.}\hfill & & & & & & \\ \phantom{\rule{3em}{0ex}}\text{Check the middle term. Is it}\phantom{\rule{0.2em}{0ex}}2ab?\hfill & & & \hfill {\left(a\right)}^{2}{}_{\text{\u2198}}\underset{2\xb7a\xb7b}{}{}_{\text{\u2199}}{\left(b\right)}^{2}\hfill & & & \hfill {\left(a\right)}^{2}{}_{\text{\u2198}}\underset{2\xb7a\xb7b}{}{}_{\text{\u2199}}{\left(b\right)}^{2}\hfill \\ \text{Step 2.}\phantom{\rule{0.2em}{0ex}}\text{Write the square of the binomial.}\hfill & & & \hfill {\left(a+b\right)}^{2}\hfill & & & \hfill {\left(a-b\right)}^{2}\hfill \\ \text{Step 3.}\phantom{\rule{0.2em}{0ex}}\text{Check by multiplying.}\hfill & & & & & & \end{array}$**Factor differences of squares**See Example 7.47.

$\begin{array}{cccc}\text{Step 1.}\phantom{\rule{0.2em}{0ex}}\text{Does the binomial fit the pattern?}\hfill & & & \hfill {a}^{2}-{b}^{2}\hfill \\ \phantom{\rule{3em}{0ex}}\text{Is this a difference?}\hfill & & & \hfill \_\_\_\_-\_\_\_\_\hfill \\ \phantom{\rule{3em}{0ex}}\text{Are the first and last terms perfect squares?}\hfill & & & \\ \text{Step 2.}\phantom{\rule{0.2em}{0ex}}\text{Write them as squares.}\hfill & & & \hfill {\left(a\right)}^{2}-{\left(b\right)}^{2}\hfill \\ \text{Step 3.}\phantom{\rule{0.2em}{0ex}}\text{Write the product of conjugates.}\hfill & & & \hfill \left(a-b\right)\left(a+b\right)\hfill \\ \text{Step 4.}\phantom{\rule{0.2em}{0ex}}\text{Check by multiplying.}\hfill & & & \end{array}$**Factor sum and difference of cubes**To factor the sum or difference of cubes: See Example 7.54.- 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.

## 7.5 General Strategy for Factoring Polynomials

**General Strategy for Factoring Polynomials See Figure 7.4.****How to Factor 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.

- 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.

- If it is a binomial:
- Step 3. Check. Is it factored completely? Do the factors multiply back to the original polynomial?

## 7.6 Quadratic Equations

**Zero Product Property**If $a\xb7b=0$, then either $a=0$ or $b=0$ or both. See Example 7.69.**Solve a quadratic equation by factoring**To solve a quadratic equation by factoring: See Example 7.73.- 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.

**Use a problem solving strategy to solve word problems**See Example 7.80.- 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 algebra equation. - Step 5.
**Solve**the equation using good 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.