Polynomial Factorization Techniques and Examples

1. Objectives Use the Factor Theorem to determine factors of a polynomial. Factor the sum and difference of two cubes.

2. 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.

3. 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)

4. Check It Out! Example 1 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)

5. Zero-Product Property Example 1:

6. Zero-Product Property Example 2: -10m3 – 2m2

7. Zero-Product Property Example 3:

8. Factoring Trinomials Example 1:

9. Factoring trinomials Example 2:

10. Factoring trinomials Example 3:

11. Factoring trinomials Example 4:

12. Factoring trinomials Example 5: 3x2 – 5x + 2

13. Determine whether the given polynomial is a factor of the polynomial P(x): • (x – 2); P(x) = 5x3 + x2 – 7 • (x – 8); P(x) = x5 – 8x4 + 8x – 64 • Use the zero-product property to solve the equation: • (t + 4)(t – 4) = 0 • (c + 7)(c + 2) = 0 • ( d + ¾)(d + 5/8) = 0 • (x + 5.4)(x – 3) = 0 • (5y – 1)(2y + 4) = 0 • (7t + 21)(t + 9) = 0 • (2n – 7)(5n + 20) = 0 • (2x – ½)(2x + ½) = 0

14. Factoring trinomials Example 6: 6x3 + 24x2 + 24x

15. Factoring trinomials Example 7: 2t3 – 2t2 – 12t

16. Factoring trinomials Example 8: 7x2 – 12x + 5

17. Factoring trinomials Example 9:

21. Example 2: Factoring by Grouping Factor: x3 – x2– 25x + 25.

22. 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. 

23. Check It Out! Example 2a Factor: x3 – 2x2– 9x + 18.

24. Check It Out! Example 2a 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. 

25. Check It Out! Example 2b Factor: 2x3 + x2+ 8x + 4.

26. 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.

27. Example 3A: Factoring the Sum or Difference of Two Cubes Factor the expression. 4x4 + 108x

28. Example 3B: Factoring the Sum or Difference of Two Cubes Factor the expression. 125d3 – 8

29. Check It Out! Example 3a Factor the expression. 8 + z6

30. Check It Out! Example 3b Factor the expression. 2x5 – 16x2

31. 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).

32. 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).

33. Lesson Quiz 1. x – 1; P(x) = 3x2 – 2x + 5 2. x+ 2; P(x) = x3 + 2x2 – x – 2 3. x3 + 3x2– 9x – 27 4. x3 + 3x2 – 28x – 60 4. 64p3 – 8q3