Lynn Marecek; Andrea Honeycutt Mathis

Key Concepts

6.1 Greatest Common Factor and Factor by Grouping

How to find the greatest common factor (GCF) of two expressions.
1. Step 1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
2. Step 2. List all factors—matching common factors in a column. In each column, circle the common factors.
3. Step 3. Bring down the common factors that all expressions share.
4. Step 4. Multiply the factors.
Distributive Property: If a, b, and c are real numbers, then

$a (b + c) = a b + a c and a b + a c = a (b + c)$

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.
1. Step 1. Find the GCF of all the terms of the polynomial.
2. Step 2. Rewrite each term as a product using the GCF.
3. Step 3. Use the “reverse” Distributive Property to factor the expression.
4. Step 4. Check by multiplying the factors.
Factor as a Noun and a Verb: We use “factor” as both a noun and a verb.

$\begin{matrix} Noun: & 7 is a factor of 14 \\ Verb: & factor 3 from 3 a + 3 \end{matrix}$
How to factor by grouping.
1. Step 1. Group terms with common factors.
2. Step 2. Factor out the common factor in each group.
3. Step 3. Factor the common factor from the expression.
4. Step 4. Check by multiplying the factors.

6.2 Factor Trinomials

How to factor trinomials of the form $x^{2} + b x + c .$
1. Step 1. Write the factors as two binomials with first terms x. $\begin{matrix} x^{2} + b x + c \\ (x) (x) \end{matrix}$
2. Step 2. Find two numbers m and n that
  $\begin{matrix} multiply to & c, m \cdot n = c \\ add to & b, m + n = b \end{matrix}$
3. Step 3. Use m and n as the last terms of the factors. $(x + m) (x + n)$
4. Step 4. Check by multiplying the factors.
Strategy for Factoring Trinomials of the Form $x^{2} + b x + c$ : When we factor a trinomial, we look at the signs of its terms first to determine the signs of the binomial factors.
$\begin{matrix} \begin{matrix} x^{2} + b x + c \\ (x + m) (x + n) \end{matrix} \\ When c is positive, m and n have the same sign. \\ b positive b negative \\ m, n positive m, n negative \\ x^{2} + 5 x + 6 x^{2} - 6 x + 8 \\ (x + 2) (x + 3) (x - 4) (x - 2) \\ same signs same signs \\ When c is negative, m and n have opposite signs. \\ x^{2} + x - 12 x^{2} - 2 x - 15 \\ (x + 4) (x - 3) (x - 5) (x + 3) \\ opposite signs opposite signs \end{matrix}$
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 $a x^{2} + b x + c$ using trial and error.
1. Step 1. Write the trinomial in descending order of degrees as needed.
2. Step 2. Factor any GCF.
3. Step 3. Find all the factor pairs of the first term.
4. Step 4. Find all the factor pairs of the third term.
5. Step 5. Test all the possible combinations of the factors until the correct product is found.
6. Step 6. Check by multiplying.
How to factor trinomials of the form $a x^{2} + b x + c$ using the “ac” method.
1. Step 1. Factor any GCF.
2. Step 2. Find the product ac.
3. Step 3. Find two numbers m and n that:
  $\begin{matrix} Multiply to a c . & m \cdot n = a \cdot c \\ Add to b . & m + n = b \\ a x^{2} + b x + c \end{matrix}$
4. Step 4. Split the middle term using m and n. $a x^{2} + m x + n x + c$
5. Step 5. Factor by grouping.
6. Step 6. Check by multiplying the factors.

6.3 Factor Special Products

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

$\begin{matrix} a^{2} + 2 a b + b^{2} = {(a + b)}^{2} \\ a^{2} - 2 a b + b^{2} = {(a - b)}^{2} \end{matrix}$
How to factor perfect square trinomials.
$\begin{matrix} Step 1. & Does the trinomial fit the pattern? & a^{2} + 2 a b + b^{2} & a^{2} - 2 a b + b^{2} \\ Is the first term a perfect square? & {(a)}^{2} & {(a)}^{2} \\ Write it as a square. \\ Is the last term a perfect square? & {(a)}^{2} {(b)}^{2} & {(a)}^{2} {(b)}^{2} \\ Write it as a square. \\ Check the middle term. Is it 2 a b ? & {(a)}^{2}_{↘} \underset{2 \cdot a \cdot b}{}_{↙} {(b)}^{2} & {(a)}^{2}_{↘} \underset{2 \cdot a \cdot b}{}_{↙} {(b)}^{2} \\ Step 2. & Write the square of the binomial. & {(a + b)}^{2} & {(a - b)}^{2} \\ Step 3. & Check by multiplying. \end{matrix}$
Difference of Squares Pattern: If $a, b$ are real numbers,
How to factor differences of squares.
$\begin{matrix} Step 1. & Does the binomial fit the pattern? & a^{2} - b^{2} \\ Is this a difference? & ____-____ \\ Are the first and last terms perfect squares? \\ Step 2. & Write them as squares. & {(a)}^{2} - {(b)}^{2} \\ Step 3. & Write the product of conjugates. & (a - b) (a + b) \\ Step 4. & Check by multiplying. \end{matrix}$
Sum and Difference of Cubes Pattern
$\begin{matrix} a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2}) \\ a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2}) \end{matrix}$
How to factor the sum or difference of cubes.
1. 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?
2. Step 2. Write them as cubes.
3. Step 3. Use either the sum or difference of cubes pattern.
4. Step 4. Simplify inside the parentheses
5. Step 5. Check by multiplying the factors.

6.4 General Strategy for Factoring Polynomials

This chart shows the general strategies for factoring polynomials. It shows ways to find GCF of binomials, trinomials and polynomials with more than 3 terms. For binomials, we have difference of squares: a squared minus b squared equals a minus b, a plus b; sum of squares do not factor; sub of cubes: a cubed plus b cubed equals open parentheses a plus b close parentheses open parentheses a squared minus ab plus b squared close parentheses; difference of cubes: a cubed minus b cubed equals open parentheses a minus b close parentheses open parentheses a squared plus ab plus b squared close parentheses. For trinomials, we have x squared plus bx plus c where we put x as a term in each factor and we have a squared plus bx plus c. Here, if a and c are squares, we have a plus b whole squared equals a squared plus 2 ab plus b squared and a minus b whole squared equals a squared minus 2 ab plus b squared. If a and c are not squares, we use the ac method. For polynomials with more than 3 terms, we use grouping.

How to use a general strategy for factoring polynomials.
1. Step 1. Is there a greatest common factor?
  Factor it out.
2. 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 $x^{2} + b x + c ?$ Undo FOIL.
  Is it of the form $a x^{2} + b x + c ?$
  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.
3. Step 3. Check.
  Is it factored completely?
  Do the factors multiply back to the original polynomial?

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 $a x^{2} + b x + c = 0$ is called a quadratic equation.

$a, b, c are real numbers and a \neq 0$
Zero Product Property: If $a \cdot b = 0,$ then either $a = 0$ or $b = 0$ or both.
How to use the Zero Product Property
1. Step 1. Set each factor equal to zero.
2. Step 2. Solve the linear equations.
3. Step 3. Check.
How to solve a quadratic equation by factoring.
1. Step 1. Write the quadratic equation in standard form, $a x^{2} + b x + c = 0 .$
2. Step 2. Factor the quadratic expression.
3. Step 3. Use the Zero Product Property.
4. Step 4. Solve the linear equations.
5. Step 5. Check. Substitute each solution separately into the original equation.
Zero of a Function: For any function f, if $f (x) = 0,$ then x is a zero of the function.
How to use a problem solving strategy to solve word problems.
1. Step 1. Read the problem. Make sure all the words and ideas are understood.
2. Step 2. Identify what we are looking for.
3. Step 3. Name what we are looking for. Choose a variable to represent that quantity.
4. 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.
5. Step 5. Solve the equation using appropriate algebra techniques.
6. Step 6. Check the answer in the problem and make sure it makes sense.
7. Step 7. Answer the question with a complete sentence.