4.6 Factoring Polynomials Objective • review factoring quadratics • factor by grouping. • factor perfect square binomials • factor difference of squares • factor difference of cubes • factor sum of cubes

Factoring Chart This chart will help you to determine which method of factoring to use. Type Number of Terms 1. GCF 2 or more 2. Diff. Of Squares 2 Trinomials 3 Sum-diff of cubes 2

Concept: Review Factoring Trinomials Factor. 3x2 + 14x + 8 This time, the x2 term DOES have a coefficient (other than 1)! Step 1: Multiply 3 • 8 = 24 (the leading coefficient & constant). 24 = 1 • 24 = 2 • 12 = 3 • 8 = 4 • 6 Step 2: List all pairs of numbers that multiply to equal that product, 24. Step 3: Which pair adds up to 14?

Concept: Factoring Trinomials cont… Factor. 3x2 + 14x + 8 Step 4: Write temporary factors with the two numbers. ( x + )( x + ) 2 12 3 3 Step 5: Put the original leading coefficient (3) under both numbers. 4 2 ( x + )( x + ) 12 3 Step 6: Reduce the fractions, if possible. 2 ( x + )( x + ) 4 3 ( 3x + 2 )( x + 4 ) Step 7: Move denominators in front of x.

Concept: Factoring Trinomials cont… Factor. 3x2 + 14x + 8 You should always check the factors by distributing, especially since this process has more than a couple of steps. ( 3x + 2 )( x + 4 ) = 3x • x + 3x • 4 + 2 • x + 2 • 4 = 3x2 + 14 x + 8 √ 3x2 + 14x + 8 = (3x + 2)(x + 4)

Concept: You Try Factor: 2x2 + 11x + 12 Answer: (2x + 3)(x + 4)

Concept: Difference of Squares Given: (Ax2 – B) where all given terms are perfect squares, you can quickly factor the binomial as follows: (Ax2 – B) = (√Ax2 + √B) (√Ax2 – √B) Ex: 9x2 – 25 (√9x2 + √25) (√9x2 – √25) ( ) ( ) 3x + 5 3x – 5

Concept: You Try!!! 16x2 – 4 Answer: (4x + 2)(4x – 2)

Concept: Factoring By Grouping Given a polynomial in the form: ax3 + bx2 + cx + d Check to see if you can factor out a common term Group in pairs so they have a common factor. 2x4 + 8x3 – 18x2 – 72x 2x 2x 2x 2x 2x[x3 + 4x2 – 9x –36] 2x[(x3 + 4x2) + (– 9x –36)]

Concept: Factoring Other Powers 2x[(x3 + 4x2) + (– 9x –36)] factor: x2 –9 2x[ x2(x + 4) –9(x + 4) ] 2x(x2 – 9)(x + 4) 3) Factor out the common term from each group. Notice the green ( ) are the same. Write as two groupings.

Concept: Factoring Other Powers 2x (x2 – 9)(x + 4) (x2 – 9) is a difference of squares and will factor apart. (x2 – 9) (x + 3)(x – 3) Answer: 2x(x + 3)(x – 3)(x + 4) Check to see if either of the two sets of ( ) will factor any further. 6) Write all of the factors.

Concept: You Try Factor: 3x3 + 6x2 – 3x – 6 2) 4x3 + 4x2 + 28x + 28 Answers: 3(x+1)(x–1)(x+2) 4(x2 + 7)(x+1) Note: this is not a difference of squares