Section 5.5 Notes: Solving Polynomial Equations

S.O.A.P. = Same Opposite Always Positive Polynomials that cannot be factored are called prime polynomials. Perfect Cube: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 (you will want to be familiar with these).

Example 1: Factor each polynomial Example 1: Factor each polynomial. If the polynomial cannot be factored, write prime. a) x3 + 125 a3: x3 b3: 125 a: x b: 5 a2: x2 ab: 5x b2: 25 (a + b)(a2 – ab + b2) = (x + 5)(x2 – 5x + 25)

Example 1: Factor each polynomial Example 1: Factor each polynomial. If the polynomial cannot be factored, write prime. b) 8y3 – 27 a3: 8y3 b3: 27 a: 2y b: 3 a2: 4y2 ab: 6y b2: 9 (a – b)(a2 + ab + b2) = (2y – 3)(4y2 + 6y + 9)

Example 1 cont. : Factor each polynomial Example 1 cont.: Factor each polynomial. If the polynomial cannot be factored, write prime. c) 64h4 – 27h GCF: h a3: 64h3 b3: 27 a: 4h b: 3 a2: 16h2 ab: 12h b2: 9 (a – b)(a2 + ab + b2) = (4h – 3)(16h2 + 12h + 9) Don’t forget about your GCF: FINAL ANSWER: h(4h – 3)(16h2 + 12h + 9) d) x3y + 343y

Example 1 cont. : Factor each polynomial Example 1 cont.: Factor each polynomial. If the polynomial cannot be factored, write prime. d) x3y + 343y GCF: y a3: x3 b3: 343 a: x b: 7 a2: x2 ab: 7x b2: 49 (a + b)(a2 – ab + b2) = (x + 7)(x2 – 7x + 49) Don’t forget about your GCF: FINAL ANSWER: y(x + 7)(x2 – 7x + 49)

1. Split the terms in a polynomial into two groups 1. Split the terms in a polynomial into two groups. (x3 and x2), then (x and #) 2. Pull out a GCF from each group, so the remaining factors are the same. 3. Make 2 factors. (gcfs and parentheses) 4. Factor the factors if possible.

Example 2: Factor each polynomial Example 2: Factor each polynomial. If the polynomial cannot be factored, write prime. a) 2x3 – 3x2 – 10x + 15 (2x3 – 3x2)( – 10x + 15) GCF of first set: x2 GCF of second set: –5 x2(2x – 3) –5(2x – 3) Notice that the parentheses are the same Final Answer: (2x – 3)(x2 – 5) b) x2 y2 – 3x2 – 4y2 + 12

Example 2: Factor each polynomial Example 2: Factor each polynomial. If the polynomial cannot be factored, write prime. b) x2 y2 – 3x2 – 4y2 + 12 (x2 y2 – 3x2) (– 4y2 + 12) GCF of first set: x2 GCF of second set: –4 x2(y2 – 3) –4(y2 – 3) Notice that the parentheses are the same (y2 – 3)(y2 – 4) (please note that y2 – 4 is a difference of two sqaures) Final Answer: (y2 – 3)(y + 2)(y – 2)

Example 2: Factor each polynomial Example 2: Factor each polynomial. If the polynomial cannot be factored, write prime. c) ax2 + 2a + 2b + bx2 (ax2 + 2a) ( + 2b + bx2) GCF of first set: a GCF of second set: b a(x2 + 2) + b(2 + x2) Notice that the parentheses are the same even though they are in different order, just change it around a(x2 + 2) + b(x2+ 2) Final Answer: (x2+ 2)(a + b)