Multiply: a) x ( x + 7) b) 6 x ( x 2  4 x + 5) Solution

2. Example Multiply: a) x(x + 7) b) 6x(x2 4x + 5) Solution a) x(x + 7) = x x + x  7 = x2 + 7x b) 6x(x2 4x + 5) = (6x)(x2)  (6x)(4x) + (6x)(5) = 6x3  24x2 + 30x

4. L F I O Example Multiply: (x + 4)(x2+ 3). Solution F O I L (x + 4)(x2+ 3) = x3 + 3x + 4x2 + 12 = x3 + 4x2 + 3x + 12 The terms are rearranged in descending order for the final answer. Example Multiply. a) (x + 8)(x+ 5) b) (y + 4) (y 3) c) (5t3 + 4t)(2t2  1) d) (4  3x)(8  5x3) Solution a) (x + 8)(x+ 5) = x2 + 5x + 8x + 40 = x2 + 13x + 40 b) (y + 4) (y 3) = y2  3y + 4y  12 = y2 + y  12

5. Example continued Solution c) (5t3 + 4t)(2t2  1) = 10t5 5t3 + 8t3  4t = 10t5 + 3t3 4t d) (4  3x)(8  5x3) = 32  20x3  24x + 15x4 = 32  24x  20x3 + 15x4 In general, if the original binomials are written in ascending order, the answer is also written that way.

6. Example Multiply. a) (x + 8)2 b) (y 7)2 c) (4x  3x5)2 Solution a) (x + 8)2 = x2 + 2x8 + 82 = x2 + 16x + 64

7. Example continued Solution (A – B)2 = A2 2AB + B2 b) (y 7)2 = y2  2  y  7 + 72 = y2  14y + 49 c) (4x  3x5)2 = (4x)2  2  4x  3x5 + (3x5)2 = 16x2  24x6 + 9x10

8. Multiply. a) (x + 8)(x 8) b) (6 + 5w) (6  5w) c) (4t3  3)(4t3 + 3) Solution a) (x + 8)(x 8) = x2 82 = x2  64 c) (4t3  3)(4t3 + 3) = (4t3)2  32 = 16t6 9 b) (6 + 5w) (6  5w) = 62  (5w)2 = 36  25w2

9. Examples Function Notation • Given f(x) = x2 – 6x + 7, find and simplify each of the following. • a) f(a) + 3 b) f(a + 3) • Solution • a) To find f(a) + 3, we replace x with a to find f(a). Then we add 3 to the result. • f(a) + 3 = a2 – 6a + 7 + 3 • = a2 – 6a + 10 b) To find f(a + 3), we replace x with a + 3 f(a + 3) = (a + 3)2 – 6(a + 3) + 7 = a2 + 6a + 9 – 6a – 18 + 7 = a2 – 2

10. Examples