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Factoring Polynomials

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Presentation transcript:

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 – x  3 + 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)

Check It Out! Example 3a Factor the expression. 8 + z6 Rewrite as the difference of cubes. (2)3 + (z2)3 (2 + z2)[(2)2 – 2  z^2 + (z2)2] Use the rule a3 + b3 = (a + b)  (a2 – ab + b2). (2 + z2)(4 – 2z^2 + z4)

Check It Out! Example 3b 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 + x  2 + 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 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.

Check It Out! Example 4 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.

Check It Out! Example 4 Continued One corresponding factor is (x – 1). 1 1 –8 19 –12 Use synthetic division to factor the polynomial. 1 –7 12 1 –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.