Unit 3.1 Rational Expressions, Equations, and Inequalities Intro: Factoring Review Objectives: To factor a monomial from a polynomial To factor trinomials of the form ax2 + bx + c To factor higher degree polynomials by grouping To factor sum or difference of cubes

Finding the Greatest Common Factor GCF - The highest number that divides exactly into two or more numbers. 

Example 1: Finding the Greatest Common Factor -x5 + 9x3 5x3 + 25x2 + 45x List the prime factors of each term. Identify the factors common to all terms. Are there any common factors? How about -1?

Factoring by Grouping First group the two terms with the highest degrees. If that doesn’t work, try another grouping. Your goal is to find a common binomial factor 

Example 2: Factoring by Grouping 3n3 – 12n2+ 2n – 8 8x3 + 14x2 + 20x + 35 Factor out the GCF of each group of two terms

Factoring Quadratic Trinomials when a = 1. ax2 + bx + c To factor a trinomial of the form ax2 + bx + c as a product of binomials: You want the factors in the form (x + m) (x + n) for some numbers m and n. You will want to find two numbers that mult. to equal the constant term ‘c’ and also add to equal ‘b’

Example 3: Factoring Quadratic Trinomials when a = 1. ax2 + bx + c This polynomial appears to be missing the “bx” term. Rewrite the polynomial with a coefficient of 0 with the variable x. Identify the pair of factors of -16 that have a sum of 0. SAME FACTORS _____ × _____ = -16 _____ + _____ = 0 Identify the pair of factors of 24 that have a sum of 11. SAME FACTORS _____ × _____ = 24 _____ + _____ = 11 x2 + 11x + 24 x2 + 2x – 15 x2 – 16 Identify the pair of factors of -15 that have a sum of 2. SAME FACTORS _____ × _____ = -15 _____ + _____ = 2

Factoring Quadratic Trinomials when a ≠ 1. ax2 + bx + c To factor a trinomial of the form ax2 + bx + c Find two numbers that mult. to equal ‘ac’ and add to equal ‘b’ Rewrite the middle term ‘bx’ with the new terms Factor the first two and last two terms separately (factor by grouping) If you have done this correctly, our two new terms should clearly have a common factor.

Example 4: Factoring Quadratic Trinomials when a ≠ 1. ax2 + bx + c 5x2 + 11x + 2 2x2 - 13x - 7 Step 1: Find factors of ac that have a sum of b. Since ac = 10 and b = 11, find positive factors of 10 that have sum 11 _______ × _______ = 10 _______ + _______ = 11 Step 1: Find factors of ac that have a sum of b. Since ac = -14 and b = -13, find factors of -14 that have sum -13 _______ × _______ = -14 _______ + _______ = -13 Step 2: To factor the trinomial, use the factors you found to rewrite bx (order is not important) Step 3: Factor the first two and last two terms separately (if we have done this correctly, our two new terms should have a common factor)

1st check for GCF – Factor (divide) out the greatest common factor If the leading coefficient is negative, factor out a -1 with the GCF 3 terms Quadratic Trinomial ax2 + bx +c when a = 1 3 terms Quadratic Trinomial ax2 + bx +c when a ≠ 1 4 terms Grouping