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Polynomials and Polynomial Functions

Learn how to factor trinomials using the reverse multiplication method. Find the factors and factor out the common factors to simplify the trinomials. Practice examples included.

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Polynomials and Polynomial Functions

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  1. Chapter 5 Polynomials and Polynomial Functions

  2. Chapter Sections 5.1 – Addition and Subtraction of Polynomials 5.2 – Multiplication of Polynomials 5.3 – Division of Polynomials and Synthetic Division 5.4 – Factoring a Monomial from a Polynomial and Factoring by Grouping 5.5 – Factoring Trinomials 5.6 – Special Factoring Formulas 5.7-A General Review of Factoring 5.8- Polynomial Equations

  3. Factoring Trinomials § 5.5

  4. Factoring Trinomials Recall that factoring is the reverse process of multiplication. Using the FOIL method, we can show that (x – 3)(x – 8) = x2 – 11x + 24. Therefore x2 – 11x + 24 = (x – 3)(x – 8). Note that this trinomial results in the product of two binomials whose first term is x and second term is a number (including its sign).

  5. L x2 + bx + c = (x +?)(x +?) F F O I L = 14x2 + 28x + 6x + 12 = 14x2 + 34x + 12 (7x + 3)(2x + 4) Numbers go here. I O Factoring Trinomials Factoring any polynomial of the form x2 + bx + c will result in a pair of binomials:

  6. Factoring Trinomials • Find two numbers (or factors) whose product is c and whose sum is b. • The factors of the trinomial will be of the form (x + one number) (x + second number)

  7. Examples a.) Factor x2 - x – 12. a = 1, b = -1, c = -12. We must find two numbers whose product is c, which is -12, and whose sum is b, which is -1. We begin by listing the factors of -12, trying to find a pair whose sum is -1. continued

  8. Examples Factors of -12 (1)(-12) (2)(-6) (3)(-4) (4)(-3) (6)(-2) (12)(1) Sum of Factors 1 + (-12) = -11 2 + (-6) = -4 3 + (-4) = -1 4 + (-3) = 1 6 + (-2) = 4 12 + (-1) = 11 The numbers we are seeking are 3 and -4 because their product is -12 and their sum is -1. x2 - x – 12 = (x + 3)(x – 4)

  9. Factor out a Common Factor The first step when factoring any trinomial is to determine whether all three terms have a common factor. If so, factor out the GCF. Example Factor 3x4 – 6x3 – 72x2. The factor 3x2 is common to all three terms. Factor it out first. = 3x2(x2 – 2x – 24) We find that -6 and 4 are the factors to the trinomial in the parentheses. Therefore, 3x4 – 6x3 – 72x2 = 3x2(x – 6)(x + 4)

  10. Trial and Error Method • Write all pairs of factors of the coefficient of the squared term, a. • Write all pairs of factors of the constant, c. • Try various combinations of these factors until the correct middle term, bx, is found.

  11. Trial and Error Method Example: Factor 3t2 – 13t + 10. There is no factor common to all three terms. Next we determine that a is 3 and the only factors of 3 and 1 are 1 and 3. Therefore we write, 3t2 – 13t + 10 = (3t )(t ) Next, we look for the factors that give us the correct middle term, - 13t. Continued.

  12. Trial and Error Method Example continued: Thus, 3t2 – 13t + 10 = (3t – 10)(t – 1).

  13. Factor Trinomials of the Form ax2 + bx + c, a ≠ 1, Using Grouping To Factor Trinomials of the Form ax2 + bx + c, a ≠ 1, Using Grouping Find two numbers whose product is a · c and whose sum is b. Rewrite the middle term, bx, using the numbers found in step 1. Factor by grouping.

  14. Factor Trinomials Using Substitution Sometimes a more complicated trinomial can be factored by substituting one variable for another. Example Factor 3z4 – 17z2 – 28. Let x = z2. Then the trinomial can be written Now substitute z2 for x.

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