Lesson 3 Factoring Polynomials Skills

Transcription

1 Lesson 3 Factoring Polynomials Skills I can common factor polynomials. I can factor trinomials like where a is 1. ie. I can factor trinomials where a is not 1. ie. I can factor special products. Common Factoring Polynomial expressions can be factored in this way when each term has a greatest common factor (GCF). When we common factor we "undo" the distribution law. ie. ab + ac = a(b + c), where a is the greatest common factor. Greatest Common Factor (GCF) the largest number and the variable(s) with the greatest exponent that is common to all terms. Common Factoring Steps 1. Identify the GCF. 2. Divide each term by the GCF.

2 Common Factoring Example 1 Factor. A. B. C. Factoring Trinomials When a = 1 We must recognize a trinomial first in order to factor this type of expression. Common factoring will only take us so far. What we must recognize is a trinomial in the form where the coefficient of the squared term, a, is equal to 1. The factors of a trinomial in this form are:, where r and s are two numbers that multiply to equal c, and add to equal b.

3 Example 2 Factor. A. B. C. D. Warm Up

4 Factoring Trinomials When a 1 Again, we must recognize this trinomial as well to select our factoring strategy. What we must recognize is a trinomial in the form where the coefficient of the squared term, a, is equal some value other than 1. The factors of a trinomial in this form are:, where: p and q are numbers that multiply to equal a r and s are numbers that multiply to equal c and ps and qr add to equal b We use two methods to find these numbers p, q, r and s. 1) The Box Method 2) Decomposition or Factoring by Grouping The Box Method The box method relies on your knowledge of your multiplication tables and trial and error. It can be the shorter method when dealing with small numbers, but can be tedious and time consuming with larger numbers. Steps 1. Recognize the trinomial in the form Example Common factor if possible. 2. Draw a box with four boxes inside. 3. Numbers that multiply to give you a go in the boxes in the first column. p q = 1 x 2 4. Numbers that multiply to give you c go in the boxes in the last column. 5. The correct arrangement of numbers is determined by multiplying diagonally and adding these products together. The correct sum is the middle term, b. 6. Write the answer in the form (px r)(qx s) p q r s r s

5 Decomposition (Factoring by Grouping) The decomposition method is an algebraic procedure that is systematic (no trial and error). It can be the longer method when dealing with small numbers, but is preferential when dealing with larger numbers. Steps 1. Recognize the trinomial in the form Example Common factor if possible. 2. Multiply a by c. Look for two numbers that will multiply to give you this value, and add to give you b. 2( 15) = 30 Need two numbers that multiply to 30 and add to Rewrite term b using the two numbers you found. 4. Common factor the first two terms, then common factor the second two terms. (GROUPING) 5. Common factor the binomial in the brackets. (If you do not see the same binomial in the brackets after step 4 you made an error) Example 3 Factor. A. B. C.

6 Practice Practice Factoring Worksheet Factoring Special Products There are two types of special products we must recognize when factoring. 1) Perfect Squares 2) Differences of Squares The Perfect Square The perfect square comes from the expansion of: and

7 Example 4 Factor. A. B. C. The Difference of Squares The difference of squares comes from the expansion of: For example, look at each expansion:

8 Example 5 Factor. A. B. C. More Examples of Special Products

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10 Test Yourself! Factor, and specify in words which strategy is required. a. b. c. d. e. f.

11 Special Cases In some situations we may be required to factor by grouping and then continue to factor using strategies discussed so far (trinomials, special products, etc.)

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13 Some more fun ones! A. B. C. D. Summary When factoring always follow these steps: Common Factor (where possible) Trinomial a equals 1 a does not equal 1 Special Product perfect square difference of squares More than 3 terms (Factoring by Grouping) Practice. Section 2.3, p , # 1 9 Worksheet Extra Practice Factoring

14 Test Yourself Factor. A. B. C. D. Practice Factoring Special Products Worksheet Factoring by Grouping Worksheet Section 2.3, pages , #1 7, 9