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 Trinomials of the Form x2+bx+c

  • Factor trinomials of the form x2+bx+c

7.3 Factor Trinomials of the Form ax2+bx+c

  • Factor Trinomials of the Form ax2+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 ax2+bx+c Using the “ac” Method: See Example 7.38.
  • Choose a strategy to factor polynomials completely (updated):

7.4 Factor Special Products

  • Factor perfect square trinomials See Example 7.42.

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.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 x2+bx+c? Undo FOIL.
        Is it of the form ax2+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?

7.6 Quadratic Equations

  • Zero Product Property If ab=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, ax2+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.