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This article provides warm-up exercises and examples for factoring polynomial expressions, including finding factors, using the factor theorem, factoring the sum/difference of two cubes, and factoring by grouping.
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Warm Up Factor each expression. 1. 3x – 6y 3(x – 2y) 2. a2 – b2 (a + b)(a – b) Find each product. 3. (x – 1)(x + 3) x2+ 2x – 3 4. (a + 1)(a2 + 1) a3+ a2 + a + 1
Objectives Use the Factor Theorem to determine factors of a polynomial. Factor the sum and difference of two cubes.
Recall that if a number is divided by any of its factors, the remainder is 0. Likewise, if a polynomial is divided by any of its factors, the remainder is 0. The Remainder Theorem states that if a polynomial is divided by (x – a), the remainder is the value of the function at a. So, if (x – a) is a factor of P(x), then P(a) = 0.
Example 1: Determining Whether a Linear Binomial is a Factor Determine whether the given binomial is a factor of the polynomial P(x). A. (x + 1); (x2 – 3x + 1) B. (x + 2); (3x4 + 6x3 – 5x – 10) Find P(–1) by synthetic substitution. Find P(–2) by synthetic substitution. –1 1 –3 1 –1 4 –2 3 6 0 –5 –10 1 –4 5 –6 0 0 10 3 0 0 –5 0 P(–1) = 5 P(–1) ≠ 0, so (x + 1) is not a factor of P(x) = x2 – 3x + 1. P(–2) = 0, so (x + 2) is a factor of P(x) = 3x4 + 6x3 – 5x – 10.
Huddle Determine whether the given binomial is a factor of the polynomial P(x). b. (3x – 6); (3x4 – 6x3 + 6x2 + 3x – 30) a. (x + 2); (4x2 – 2x + 5) Find P(–2) by synthetic substitution. Divide the polynomial by 3, then find P(2) by synthetic substitution. –2 4 –2 5 –8 20 2 1 –2 2 1 –10 4 –10 25 2 0 4 10 1 0 2 5 0 P(–2) = 25 P(–2) ≠ 0, so (x + 2) is not a factor of P(x) = 4x2 – 2x + 5. P(2) = 0, so (3x – 6) is a factor of P(x) = 3x4 – 6x3 + 6x2 + 3x – 30.
You are already familiar with methods for factoring quadratic expressions. You can factor polynomials of higher degrees using many of the same methods you learned in Lesson 5-3.
Example 2: Factoring by Grouping Factor: x3 – x2– 25x + 25. (x3 – x2) + (–25x + 25) Group terms. Factor common monomials from each group. x2(x – 1) – 25(x – 1) Factor out the common binomial (x – 1). (x – 1)(x2 – 25) Factor the difference of squares. (x – 1)(x – 5)(x + 5)
Example 2 Continued Check Use the table feature of your calculator to compare the original expression and the factored form. The table shows that the original function and the factored form have the same function values.
Huddle Factor: x3 – 2x2– 9x + 18. (x3 – 2x2) + (–9x + 18) Group terms. Factor common monomials from each group. x2(x – 2) – 9(x – 2) Factor out the common binomial (x – 2). (x – 2)(x2 – 9) Factor the difference of squares. (x – 2)(x – 3)(x + 3)
Huddle Check Use the table feature of your calculator to compare the original expression and the factored form. The table shows that the original function and the factored form have the same function values.
Mastery Factor: 2x3 + x2+ 8x + 4. (2x3 + x2) + (8x + 4) Group terms. Factor common monomials from each group. x2(2x + 1) + 4(2x + 1) Factor out the common binomial (2x + 1). (2x + 1)(x2 + 4) (2x + 1)(x2 + 4)
Just as there is a special rule for factoring the difference of two squares, there are special rules for factoring the sum or difference of two cubes.
Example 3A: Factoring the Sum or Difference of Two Cubes Factor the expression. 4x4 + 108x 4x(x3 + 27) Factor out the GCF, 4x. 4x(x3 + 33) Rewrite as the sum of cubes. Use the rule a3 + b3 = (a + b) (a2– ab + b2). 4x(x + 3)(x2–x3+32) 4x(x + 3)(x2 – 3x + 9)
Example 3B: Factoring the Sum or Difference of Two Cubes Factor the expression. 125d3 – 8 Rewrite as the difference of cubes. (5d)3 – 23 (5d – 2)[(5d)2 + 5d 2 + 22] Use the rule a3– b3 = (a – b) (a2+ ab + b2). (5d – 2)(25d2 + 10d + 4)
Huddle Factor the expression. 8 + z6 Rewrite as the difference of cubes. (2)3 + (z2)3 (2+ z2)[(2)2 – 2z+ (z2)2] Use the rule a3 + b3 = (a + b) (a2– ab + b2). (2+ z2)(4 – 2z + z4)
Mastery Factor the expression. 2x5 – 16x2 2x2(x3 – 8) Factor out the GCF, 2x2. Rewrite as the difference of cubes. 2x2(x3 – 23) Use the rule a3– b3 = (a – b) (a2 + ab + b2). 2x2(x – 2)(x2+x2+22) 2x2(x – 2)(x2 + 2x + 4)
Example 4: Geometry Application The volume of a plastic storage box is modeled by the function V(x) = x3 + 6x2 + 3x – 10. Identify the values of x for which V(x) = 0, then use the graph to factor V(x). V(x) has three real zeros at x = –5, x = –2, and x = 1. If the model is accurate, the box will have no volume if x = –5, x = –2, or x = 1.
Example 4 Continued One corresponding factor is (x – 1). 1 1 6 3 –10 Use synthetic division to factor the polynomial. 1 7 10 1 0 7 10 V(x)= (x – 1)(x2 + 7x + 10) Write V(x) as a product. V(x)= (x – 1)(x+ 2)(x + 5) Factor the quadratic.
Huddle The volume of a rectangular prism is modeled by the function V(x) = x3 – 8x2 + 19x – 12, which is graphed below. Identify the values of x for which V(x) = 0, then use the graph to factor V(x). V(x) has three real zeros at x = 1, x = 3, and x = 4. If the model is accurate, the box will have no volume if x = 1, x = 3, or x = 4.
Huddle One corresponding factor is (x – 1). 1 1 –8 19 –12 Use synthetic division to factor the polynomial. 1 –7 12 1 0 –7 12 V(x)= (x – 1)(x2 – 7x + 12) Write V(x) as a product. V(x)= (x – 1)(x– 3)(x – 4) Factor the quadratic.
Lesson Quiz 1. x – 1; P(x) = 3x2 – 2x + 5 P(1) ≠ 0, so x – 1 is not a factor of P(x). 2. x+ 2; P(x) = x3 + 2x2 – x – 2 P(2) = 0, so x + 2 is a factor of P(x). 3. x3 + 3x2– 9x – 27 (x + 3)(x + 3)(x – 3) 4. x3 + 3x2 – 28x – 60 (x + 6)(x – 5)(x + 2) 4. 64p3 – 8q3 8(2p – q)(4p2 + 2pq + q2)
