Multiply: a) x(x + 7) b) 6x(x 2  4x + 5) Solution a) x(x + 7) = x  x + x  7 = x 2 + 7x b) 6x(x 2  4x + 5) = (6x)(x 2 )  (6x)(4x) + (6x)(5) = 6x 3  24x x Example

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

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

Multiply. a) (x + 8) 2 b) (y  7) 2 c) (4x  3x 5 ) 2 Solution a) (x + 8) 2 = x  x  = x x + 64 Example

Example continued Solution (A – B) 2 = A 2  2AB + B 2 b) (y  7) 2 = y 2  2  y  = y 2  14y + 49 c) (4x  3x 5 ) 2 = (4x) 2  2  4x  3x 5 + (3x 5 ) 2 = 16x 2  24x 6 + 9x 10

Multiply. a) (x + 8)(x  8) b) (6 + 5w) (6  5w)c) (4t 3  3)(4t 3 + 3) Solution a) (x + 8)(x  8) = x 2  8 2 = x 2  64 b) (6 + 5w) (6  5w) = 6 2  (5w) 2 = 36  25w 2 c) (4t 3  3)(4t 3 + 3) = (4t 3 ) 2  3 2 = 16t 6  9

Function Notation Given f(x) = x 2 – 6x + 7, find and simplify each of the following. a) f(a) + 3b) 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 = a 2 – 6a = a 2 – 6a + 10 Examples b) To find f(a + 3), we replace x with a + 3 f(a + 3) = (a + 3) 2 – 6(a + 3) + 7 = a 2 + 6a + 9 – 6a – = a 2 – 2

Examples