140 likes | 382 Views
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.
E N D
Chapter 6 Factoring Polynomials Section 1 Greatest Common Factor andFactoring 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 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.
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.
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.
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).
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.
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.
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
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
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)
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.
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:
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