1. ALGEBRA II HONORS @ REMAINDER and FACTOR THEOREMS

2. 1) Factor completely. a) 2x 2 – 5x - 12 ANSWER : (2x + 3)(x – 4) b) x 2 - 36 ANSWER : (x + 6)(x – 6) c) x 3 – 2x 2 – 4x + 8 ANSWER : (x + 2)(x – 2) 2

3. d) x 3 – 6x 2 – x + 30 It looks like we need another method for factoring. Take out your and type 2x 2 – 5x – 12 into y 1. Hit graph. What are the x-intercepts?ANSWER : 4, -3/2 Remember that 2x 2 – 5x – 12 factored to (2x + 3)(x - 4). Set each factor equal to zero. What happens? 2x + 3 = 0 2x = -3 x = -3/2 x – 4 = 0 x = 4

4. Now, type x 2 – 36 into y 1 and graph. What do you notice about the x-intercepts? x = 6 x – 6 = 0 x = -6 x + 6 = 0 Notice that x 2 – 36 factors to (x – 6)(x + 6). Type x 3 – 6x 2 – x + 30 into y 1 and graph. What are the x-intercepts? SOLUTION : x-intercepts are -2, 3, and 5 The zeroes are : x = -2x = 3x = 5 x + 2 = 0x – 3 = 0x – 5 = 0 Therefore, the factors are (x + 2)(x – 3)(x – 5)

5. 2) Find the dimensions of Tyler’s house if its volume is V(x) = 2x 3 + 3x 2 – 18x + 8 m 3. That is, find its factors. SOLUTION : Type 2x 3 + 3x 2 – 18x + 8 into y 1 and graph. You can see that -4 and 2 are zeroes for V(x). But there are 3 zeroes. How can we find the other zero? ANSWER : synthetic division -423-188 -8 20 -8 22-5 20 4 -2 2-1 0 FACTORS : 2x – 1 x = -4 x + 4 = 0 x = 2 x – 2 = 0 (2x – 1)(x + 4)(x – 2)

6. REMAINDER THEOREOM : REMAINDER THEOREOM : means you will always get a remainder, even if it is zero, when you divide a polynomial by another polynomial. FACTOR THEOREM : FACTOR THEOREM : If, then ax – b is a factor of P(x) and is a root of P(x). Note : If ax – b is a factor, then ax – b = 0 ax = b x = RATIONAL ROOT THEOREM : RATIONAL ROOT THEOREM : ax – b is a factor of P(x) iff

7. 3)Find the zeroes and factors. a) P(x) = 6x 3 – 31x 2 + 25x + 12 ANSWERS :zeroes factors (x – 4)(2x – 3)(3x + 1) b) P(x) = x 4 – 7x 3 + 18x 2 – 20x + 8 ANSWERS :zeroes {1, 2} factors (x – 1)(x – 2) 3