2. Quotient Dividend Remainder Divisor Long Division

3. Long Division - A Review 8 4 6 3 1 5 4 0 Divisor Remainder Dividend Quotient 7 8 6 5 6 7 6 4 7 3 1 4 3 8 4 96

4. Divide: (x 2 + 7x + 2) ÷ (x + 2) 1. The polynomial must be in descending order of powers. Any missing terms are to be filled with a zero placeholder. x 2 + 7x + 2 x + 2 2. Only the first term is used when doing the division. Divide x 2 x x 3. Multiply your answer with the entire divisor. x(x + 2) = x 2 + 2x x 2 + 2x 4. Subtract, bring down the next term and repeat the process. 5x5x + 5 5x + 10 -8 = x + 2 multiply Division by a Binomial -( ), x  -2

5. 4x 3 - 11x 2 + 8x + 10x - 2 4x24x2 4x 3 - 8x 2 - 3x 2 - 3x - 3x 2 + 6x 2x2x + 2 2x - 4 14 + 8x + 10 Division by a Binomial NPV’s

6. x 3 + 0x 2 - 20x + 8x + 4 x2x2 x 3 + 4x 2 - 4x 2 - 4x - 4x 2 - 16x -4x - 4 -4x - 16 24 - 20x + 8 NPV’s

7. Divide x 3 - 2x 2 - 33x + 90 by (x - 5) using synthetic division. -51-2-3390 1. Write only the constant term of the divisor, and the coefficients of the dividend. 2. Bring down the first term of the dividend. 1 3. Multiply 1 by -5, record the product and subtract. -5 Multiply 3 4. Multiply 3 by -5, record the product and subtract. -15 -18 5. Multiply -18 by -5, record the product and subtract. subtract 90 0 Quotient Rem Written as x 2 + 3x - 18 Using the division statement: P(x) = (x - 5)(x 2 + 3x - 18) Synthetic Division

8. Divide: (x 4 - 2x 3 + x 2 + 12x - 6) ÷ (x - 2) -21 -2 1 12 -6 1 -2 0 0 1 14 -28 22 (x - 2)(x 3 + x + 14)+ 22 Using Synthetic Division x 4 - 2x 3 + x 2 + 12x – 6 =

9. Given P(x) = x 3 - 4x 2 + 5x + 1, determine the remainder when P(x) is divided by x - 1. 1 -4 5 1 1 -3 3 2 -2 3 The remainder is 3. Using f(x) = x 3 - 4x 2 + 5x + 1, determine P(1): P(1) = (1) 3 - 4(1) 2 + 5(1) + 1 = 1 - 4 + 5 + 1 = 3 NOTE: P(1) gives the same answer as the remainder using synthetic division. Therefore P(1) is equal to the remainder. In other words, when the polynomial x 3 - 4x 2 + 5x + 1 is divided by x - 1, the remainder is P (1). The Remainder Theorem

10. Remainder Theorem: When a polynomial P(x) is divided by x - a, the remainder is P(a). [think x - a, then x = a] Determine the remainder when x 3 - 4x 2 + 5x - 1 is divided by: a) x - 2b) x + 1 Calculate P(2) P(2) = (2) 3 - 4(2) 2 + 5(2) - 1 = 8 - 16 + 10 - 1 = 1 The remainder is 1. Calculate P(-1) P(-1) = (-1) 3 - 4(-1) 2 + 5(-1) - 1 = -1 - 4 - 5 - 1 = -11 The remainder is -11. Point (2, 1) is on the graph of of f(x) = x 3 - 4x 2 + 5x - 1 Point (-1, -11) is on the graph of of f(x) = x 3 - 4x 2 + 5x - 1

11. Determine the value of k. Whenthe remainder is 30.is divided by

12. When the polynomial 3x 3 + ax 2 + bx -9 is divided by x - 2, the remainder is -5. When the polynomial is divided by x + 1, the remainder is -16. What are the values of a and b?

13. Page 124 1, 3c,f, 4a,c, 6a,7b, 8a,c, 9, 11, 14

14. 1. (4x 3 - 11x 2 + 8x + 6) ÷ (x - 2) -2 4 -11 8 6 4 - 8 -3 6 2 -4 10 P(x) = (x - 2)(4x 2 - 3x + 2) + 10 2. (2x 3 - 2x 2 + 3x + 3) ÷ (x - 1) -1 2 -2 3 3 -2 0 -3 2 0 3 6 P(x) = (x - 1)(2x 2 + 3) + 6 Using Synthetic Division