DIVIDING POLYNOMIALS Synthetically! Mr. Richard must have your “Un-Divided” attention for this lesson!

DIVIDING POLYNOMIALS Synthetically! 1. Put the opposite value of “a” Outside the Division Problem Bring down the first number. Multiply the first number times “a” Add the numbers. Repeat the cycle. Remember: Multiply and then Add

Example: Divide 3x2 + 2x – 1 by x – 2 using synthetic division. Synthetic division is a shorter method of dividing polynomials. This method can be used only when the divisor is of the form x – a. It uses the coefficients of each term in the dividend. Example: Divide 3x2 + 2x – 1 by x – 2 using synthetic division. Since the divisor is x – 2, a = 2. value of a coefficients of the dividend 1. Bring down 3 2 3 2 – 1 2. (2 • 3) = 6 3. (2 + 6) = 8 6 16 4. (2 • 8) = 16 3 8 15 5. (–1 + 16) = 15 coefficients of quotient remainder 3x + 8 Answer: 15 Synthetic Division

Example: Synthetic Division Example: Divide x3 – 3x + 4 by x + 3 using synthetic division. Since, x – a = x + 3, a = – 3. coefficients of dividend a – 3 1 0 – 3 4 Insert zero coefficient as placeholder for the missing x2 term. – 3 9 – 18 1 – 3 6 – 14 remainder coefficients of quotient = x2 – 3x + 6 – 14 Notice that the degree of the first term of the quotient is one less than the degree of the first term of the dividend. Example: Synthetic Division

The result is 2x2 + x – 4 Ex. 1: Divide 2x3 – 7x2 – 8x + 16 by x - 4 4 2 -7 -8 16 8 4 -16 2 1 -4 The result is 2x2 + x – 4

- + + - - + 1 3x2 -2x +1 3x3 – 6x2 – 2x2 +5x -2x2 + 4x x – 1 x – 2 Let’s compare the process of synthetic division to long division. We have used both methods to divide 3x3 – 8x2 + 5x -1 by x – 2. 3x2 -2x +1 3x3 – 8x2 + 5x -1 2 3 -8 5 -1 6 -4 2 3x3 – 6x2 - + 3 -2 1 1 – 2x2 +5x -2x2 + 4x + - Compare the numbers in the second row of the synthetic division with those that appear in the long division. Why do you think that in synthetic division you add these numbers that you would subtract when using long division? Look at the divisors. x – 1 x – 2 - + 1

Synthetic Division contd.

Example

Ex. 4: Use synthetic division to find (3y3 + 2y2 – 32y + 2)  (y – 3) 3 2 -32 2 9 33 3 3 11 1 5 The result is 3y2 +11y +1 +

Ex. 3: Use synthetic division to find (x3 + 13x2 – 12x – 8)  (x + 2) -2 1 13 -12 -8 -2 -22 68 1 11 -34 60 The result is x2 +11x – 34 +

The remainder is 68 at x = 3, so f (3) = 68. Remainder Theorem: The remainder of the division of a polynomial f (x) by x – a is f (a). Example: Using the remainder theorem, evaluate f(x) = x 4 – 4x – 1 when x = 3. value of x 3 1 0 0 – 4 – 1 3 9 27 69 1 3 9 23 68 The remainder is 68 at x = 3, so f (3) = 68. You can check this using substitution: f(3) = (3)4 – 4(3) – 1 = 68. Remainder Theorem

Quiz Time! Quiz: Page 324 # 14, 16, 18, 20, 24 Homework # 13, 15, 17, 19, 21