1 / 14

Chapter 6 Factoring Polynomials

Chapter 6 Factoring Polynomials. Section 1 Greatest Common Factor and Factoring by Grouping. Section 6.1 Objectives. 1 Find the Greatest Common Factor of Two or More Expressions 2 Factor Out the Greatest Common Factor in Polynomials 3 Factor Polynomials by Grouping. factors. Factors.

carlyn
Télécharger la présentation

Chapter 6 Factoring Polynomials

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 6 Factoring Polynomials Section 1 Greatest Common Factor andFactoring by Grouping

  2. Section 6.1 Objectives 1 Find the Greatest Common Factor of Two or More Expressions 2 Factor Out the Greatest Common Factor in Polynomials 3 Factor Polynomials by Grouping

  3. factors Factors 4· 9 = 36 3(x + 2) = 3x + 6 (2x – 7)(3x + 5) = 6x2 – 11x – 35 The expressions on the left side are called factors of the expression on the right side. To factor a polynomial means to write the polynomial as a product of two or more polynomials. The greatest common factor (GCF) of a list of algebraic expressions is the largest expression that divides evenly into all the expressions.

  4. The common factors are 2 and 2. Greatest Common Factor How to Find the Greatest Common Factor of a List of Numbers Step 1: Write each number as a product of prime factors. Step 2: Determine the common prime factors. Step 3: Find the product of the common factors found in Step 2. This number is the GCF. Example: Find the GCF of 16 and 20. 16 = 2 · 2 · 2 · 2 20 = 2 · 2 · 5 The GCF is 2 · 2 = 4.

  5. The common factors are 3 and 5. Greatest Common Factor Example: Find the GCF of 60, 75, and 135. 60 = 2 · 2 · 3 · 5 75 = 3 · 5 · 5 135 = 3 · 3 · 3 · 5 The GCF is 3 · 5 = 15.

  6. The GCF as a Binomial Example: Find the GCF of 6(x – y) and 15(x – y)3. 6(x – y) = 2 · 3 · (x – y) 15(x – y)3 = 3 · 5 · (x – y) · (x – y) · (x – y) The GCF is 3 · (x – y) = 3(x – y).

  7. Finding the GCF Steps to Find the Greatest Common Factor Step 1: Find the GCF of the coefficients of each variable factor. Step 2: For each variable factor common to all terms, determine the smallest exponent that the variable factor is raised to. Step 3: Find the product of the common factors found in Steps 1 and 2. This expression is the GCF.

  8. Factoring Polynomials Steps to Factor a Polynomial Using the GCF Step 1: Identify the GCF of the terms that make up the polynomial. Step 2: Rewrite each term as the product of the GCF and the remaining factor. Step 3: Use the Distributive Property “in reverse” to factor out the GCF. Step 4: Check using the Distributive Property.

  9. Factoring Polynomials Example: Factor the trinomial 36a6 + 45a4 – 18a2 by factoring out the GCF. Step 1: Find the GCF. GCF = 9a2 Step 2: Rewrite each term as the product of the GCF and the remaining term. 36a6 + 45a4 – 18a2 = 9a2· 4a4 + 9a2 · 5a2 –9a2 · 2 Step 3: Factor out the GCF. 36a6 + 45a4 – 18a2 = 9a2(4a4 + 5a2 – 2) Step 4: Check. 9a2(4a4 + 5a2 – 2) =36a6 + 45a4 – 18a2 

  10. Factoring Out a Negative Number Example: Factor – 3x6 + 9x4 – 18x by factoring out the GCF: Step 1: Find the GCF. GCF = – 3x Step 2: Rewrite each term as the product of the GCF and the remaining term. – 3x6 + 9x4 – 18x = – 3x· x5 + (– 3x)(– 3x3) + (– 3x) · 6 Step 3: Factor out the GCF. – 3x6 + 9x4 – 18x = – 3x(x5 – 3x3 + 6) Step 4: Check. – 3x(x5 – 3x3 + 6) =– 3x6 + 9x4 – 18x 

  11. GCF = 3x + y Factoring Out a Binomial Example: Factor out the greatest common binomial factor: 6(3x + y) – z(3x + y) 6(3x + y) – z(3x + y) = (3x + y)(6 – z) Check: (3x + y)(6 – z) = 6(3x + y) – z(3x + y) 

  12. Factoring by Grouping Steps to Factor a Polynomial by Grouping Step 1: Group the terms with common factors. Step 2: In each grouping, factor out the greatest common factor. Step 3: If the remaining factor in each grouping is the same, factor it out. Step 4: Check your work by finding the product of the factors.

  13. x is the common factor. 3 is the common factor. Factor out the x from the first two terms. Factor out the 3 from the last two terms. These two factors need to be the same. Factoring by Grouping Example: Factor by grouping: x2 + 7x + 3x + 21 x2 + 7x + 3x + 21 x2 + 7x + 3x + 21 = x(x + 7) + 3(x + 7) = (x + 3)(x + 7) Factor out the common factor x + 7. (x + 3)(x + 7) = x2 + 7x + 3x + 21 Check: 

  14. Factoring by Grouping Example: Factor by grouping: xy – 4x – 3y + 12 Step 1: Group the terms with the common factors. (xy – 4x) + (– 3y + 12) Step 2: Factor out the common factor in each group. xy – 4x – 3y + 12 = x(y – 4) + (– 3)(y – 4) Step 3: Factor out the common factor that remains. = (x – 3)(y – 4) xy – 4x – 3y + 12 = (x – 3)(y – 4) Step 4: Check. (x – 3)(y – 4) = xy – 4x – 3y + 12 

More Related