Factoring Expressions This topic will be used throughout the 2 nd semester. Weekly Quizzes
What is Factoring? Quick Write: In your notes, write down everything you know about Factoring from Algebra-1 and Geometry. You can use Bullets or give examples 2 Minutes
Objectives I can factor expressions using all of these methods –Greatest Common Factor (GCF) –Difference of Two Squares –Reverse FOIL –Swing and Divide I can solve a quadratic equation using factoring
Factoring? Factors are two or more terms that multiply to form a product (Factor) x (Factor) = Product Some numbers are Prime, meaning they are only divisible by themselves and 1
Method 1 Greatest Common Factor (GCF) – the greatest factor shared by two or more terms ALWAYS try this factoring method 1 st before any other method Divide Out the Biggest common number/variable from each of the terms
Greatest Common Factors aka GCF’s Find the GCF for each set of following numbers. Find means tell what the terms have in common. Hint: list the factors and find the greatest match. a)2, 6 b)-25, -40 c)6, 18 d)16, 32 e)3, No common factors? GCF =1
Find the GCF for each set of following numbers. Hint: list the factors and find the greatest match. a)x, x 2 b)x 2, x 3 c)xy, x 2 y d)2x 3, 8x 2 e)3x 3, 6x 2 f)4x 2, 5y 3 x x2x2 xy 2x 2 Greatest Common Factors aka GCF’s 3x 2 1 No common factors? GCF =1
Factor out the GCF for each polynomial: Factor out means you need the GCF times the remaining parts. a)2x + 4y b)5a – 5b c)18x – 6y d)2m + 6mn e)5x 2 y – 10xy 2(x + 2y) 6(3x – y) 5(a – b) 5xy(x - 2) 2m(1 + 3n) Greatest Common Factors aka GCF’s How can you check?
Ex 1 15x 2 – 5x GCF = 5x 5x(3x - 1)
Ex 2 8x 2 – x GCF = x x(8x - 1)
Method #2 Difference of Two Squares a 2 – b 2 = (a + b)(a - b)
What is a Perfect Square Any term you can take the square root evenly (No decimal) x 2 y 4
Difference of Perfect Squares x 2 – 4= the answer will look like this: ( )( ) take the square root of each part: ( x 2)(x 2) Make 1 a plus and 1 a minus: (x + 2)(x - 2 )
FACTORING Difference of Perfect Squares EX: x 2 – 64 How: Take the square root of each part. One gets a + and one gets a -. Check answer by FOIL. Solution: (x – 8)(x + 8)
Example 1 (9x 2 – 16) (3x + 4)(3x – 4)
Example 2 x 2 – 16 (x + 4)(x –4)
Ex 3 36x 2 – 25 (6x + 5)(6x – 5)
Ex 4 9x PRIME
FOIL Review
Reverse FOIL When factoring a trinomial with 3 terms you will always get two factors x 2 + bx + c
Factoring a Trinomial
FOIL Method The two numbers # and * must multiply together to equal 6 The two numbers # and * must add up to 5
Example 1 Factoring Tree
Example 2 Factoring Tree
Example 3 Factoring Tree
Swing and Divide Method The Swing & Divide method is very similar to Reverse FOIL, but with 2 extra steps: You can use this method when the number in front of the x 2 term is not 1 EX: 2x 2 – x - 3
Swing & Divide 2 Steps: Swing and Divide Don’t do one without doing the 2nd
Swing & Divide Method Swing Now use FOIL Divide by Swing # Final Factors
Reducing and Checking Always reduce fractions before finding final factors Use TWO Finger Check when done
Example 2 Swing Now use FOIL Divide by Swing # Final Factors
Example 3 Swing Now use FOIL Divide by Swing # Final Factors Reduce
Board 1
Board 2
Board 3
More than ONE Method It is very possible to use more than one factoring method in a problem Remember: ALWAYS use GCF first
Example 1 2b 2 x – 50x GCF = 2x 2x(b 2 – 25) 2 nd term is the diff of 2 squares 2x(b + 5)(b - 5)
Example 2 32x 3 – 2x GCF = 2x 2x(16x 2 – 1) 2 nd term is the diff of 2 squares 2x(4x + 1)(4x - 1)
Equations To solve an equation by factoring, the equation MUST be set equal to ZERO first: ax 2 + bx + c = 0
Zero Product Property If ab = 0, then a = 0 b = 0
Solving with Factoring Set each variable factor equal to zero and solve for “x”
Factoring for solutions x 2 – 3x –4 = 0 (x – 4)(x + 1) = 0 Then by the Zero Product Property: (x – 4) = 0 or (x + 1) = 0 If we solve these for “x” we get the following solutions: x – 4 = 0, so x = 4 x + 1 = 0, so x = -1 These are the two solutions {-1, 4}
Example 2
Example 3
Example 4
Homework WS 5-1