**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.

**Factor trinomials of the form***x*^{2}+*bx*+*c*

**Factor Trinomials of the Form**See Example 7.33.*ax*^{2}+*bx*+*c*using Trial and Error:- 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**See Example 7.38.*ax*2+*bx*+*c*Using the “ac” Method:

**Choose a strategy to factor polynomials completely (updated):**

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

**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.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*ax*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?

**Zero Product Property**If*a*⋅*b*=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,
*ax*^{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.

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