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Polynomial Factoring Techniques and Equations Solving Guide

Learn how to factor polynomials, solve equations, and apply various factoring methods like factoring completely, in quadratic form, and by grouping. Practice solving cubic equations with examples. Develop your skills in factoring by understanding common monomials and sum/difference of cubes.

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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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