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6.4 factoring and solving polynomial equations

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6.4 factoring and solving polynomial equations

1. 1.  We already know how to factor these types of quadratic expressions: ◦ General Trinomial: 2x2 – 5x – 12 = (2x + 3)(x – 4) ◦ Perfect Square Trinomial: x2 + 10x + 25 = (x + 5)2 ◦ Difference of Two Squares: 4x2 – 9 = (2x + 3)(2x – 3) ◦ Common Monomial Factor: 6x2 + 15x = 3x(2x + 5)
2. 2.  Sum of Two Cubes: a3 + b3 = (a + b)(a2 – ab + b2) Example: x3 + 8 = (x + 2)(x2 – 2x + 4) Difference of Two Cubes: a3 – b3 = (a – b)(a2 + ab + b2) Example: 8x3 – 1 = (2x – 1)(4x2 + 2x + 1)
3. 3.  Factor each polynomial. x3 + 27 8x3 - 125
4. 4.  Factor x3 + 343
5. 5.  You can sometimes factor out a monomial, then use sum/difference of cubes. (When powers are 3 apart.) Example: ◦ 64x4 – 27x
6. 6.  Factor 3x4 – 24x
7. 7.  Sometimes we can factor by grouping pairs of terms with a common monomial factor: ra + rb + sa + sb = r(a + b) + s(a + b) = (r + s)(a + b) Example: x3 – 2x2 – 9x + 18
8. 8.  Factor the polynomial. x2y2 – 3x2 – 4y2 + 12
9. 9.  Factor x3 – 2x2 +4x - 8
10. 10.  An expression of the form au2 + bu + c where u is any expression in x is in quadratic form. We can factor these like a quadratic. Example: Factor 81x4 - 16
11. 11.  Factor 4x6 – 20x4 + 24x2 Factor x4 + 4x2 - 21
12. 12.  Factor x4 + 3x2 + 2
13. 13.  Remember the zero product property? If AB = 0 then A = 0 or B = 0 To solve, factor and set each factor equal to zero. Then solve for x. Only include real solutions. May have up to n solutions. (where n is the degree.)
14. 14.  Find the real number solutions of the equation. 2x5 + 24x = 14x3 2x5 – 18x = 0
15. 15.  A large concrete block is discovered by archaeologists with a volume of 330 cubic yards. The dimensions are x yards high by 13x – 11 yards long by 13x – 5 yards wide. What is the height?
16. 16.  You are building a bin to hold mulch for your garden. The bin will hold 162 cubic feet of mulch. The dimensions are x ft by 5x – 6 ft by 5x – 9 ft. How tall will the bin be?