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Polynomial Factoring Techniques and Equations Solving Guide

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  1. Section 2.4 Factoring polynomials

  2. Do-Now • Factor the following polynomials. • x2 + 5x + 6 • 4x2 – 1 • Solve the following equations. • 2x2 – 28x + 98 = 0 • 6x2 – x – 2 = 0

  3. Factoring Completely • A polynomial is factored completely if it is written as a product of unfactorable polynomials with integer coefficients. • Example: 4x(x2 – 9) is not factored completely because x2 – 9 can be factored into (x + 3)(x – 3).

  4. Factor the following • y3 – 4y2 – 12y • Now we are working with polynomials of degree greater than 2. What should our first step be? • Always look to see if you can factor out a common monomial. • Additional examples: • 3x3 + 30x2 + 75x • 5x5 – 80x3

  5. Factoring in quadratic form • Factor this expression: • x2 – 5x – 14 • Now try factoring….. • x6 – 5x3 – 14 • Your answer should look just like a factored quadratic equation, but with bigger exponents • Additional Examples: • 10x4 – 10 • 3x12 + 54x7 + 51x2

  6. Set up and equation and solve. • The volume of the box is 96cm3. • Write an equation to represent this situation and solve for x. • What is preventing us from solving this equation.

  7. Factor by Grouping • When all else fails, group the x3 and x2 together and the x and constant term together. Factor them separately and see if the result is an expression that has a common factor. • Examples: • x3 + 5x2 + 3x + 15 • x3 – 3x2 + 4x – 12

  8. Be careful when the middle sign is negative…. • Factor: • 27x3 + 45x2 – 3x – 5 • When grouping, rewrite as….. • (27x3 + 45x2) – (3x + 5) • Factor: • x3 – 7x2 – 9x + 63

  9. Set up and equation and solve. • The volume of the box is 96cm3. • Write an equation to represent this situation and solve for x. • Now use your factoring by grouping skills to solve the equation.

  10. Sum or Difference of Cubes

  11. = 2z2 (2z)3 – 53 EXAMPLE 2 Factor the sum or difference of two cubes Factor the polynomial completely. = x3 + 43 a. x3 + 64 Sum of two cubes = (x + 4)(x2 – 4x + 16) = 2z2(8z3 – 125) b. 16z5 – 250z2 Factor common monomial. Difference of two cubes = 2z2(2z – 5)(4z2 + 10z + 25)

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