5.5 - Long and Synthetic Division Worksheet Key 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Worksheet Key 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Worksheet Key 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Worksheet Key 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

Revised ©2013, vdang@houstonisd.org Polynomial Division Revised ©2013, vdang@houstonisd.org 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

Long Division Write the dividend in standard form, including terms with a coefficient of ZERO. Write division in the same way you would when dividing numbers. Multiply the answer by the divisor and then subtract –1 [meaning write the negative sign then distribute] Repeat process until it can not be done Leftover is remainder NOTE: Long division can be used no matter what size the divisor is. 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Elementary Review Solve 25 ÷ 3 through long division Quotient Divisor Dividend 3 x 8 = 24 Subtract from the top number Remainder 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Example 1 Divide x3 – 5x2 – 12x + 36 ÷ x – 2 using Long Division 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Example 2 Divide (x3 – 28x – 48) ÷ (x + 4) using Long Division 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Your Turn Divide (6x3 + 5x2 + 9) ÷ (2x + 3) using Long Division 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Example 3 Divide (2x2 + 3x – 4) ÷ (x – 2) using Long Division 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Example 4 Divide (2x4 + 4x3 – 5x2 + 3x – 2) ÷ (x2 + 2x – 3) using Long Division 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Example 5 Divide (12x4 – 5x2 – 3) ÷ (3x2 – 5) using Long Division 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Your Turn Divide (x4 + 3x2 + 1) ÷ (x2 – 2x + 3) using Long Division 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Identify the divisor and reverse the sign of the constant term. Write the coefficients of the polynomial in standard form. BRING DOWN the first coefficient MULTIPLY the first coefficient by the new divisor, identify the result under the next coefficient and add. Repeat the steps of multiplying and adding until the remainder is found GO BACKWARDS from the remainder and assign variables Write out the proper polynomial Check by multiplying the quotient with the divisor 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Example 6 Divide (x3 – 5x2 – 12x + 36) ÷ (x – 2) using Synthetic Division 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Example 6 Divide (x3 – 5x2 – 12x + 36) ÷ (x – 2) using Synthetic Division 2 · 1 = 2 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Example 6 Divide (x3 – 5x2 – 12x + 36) ÷ (x – 2) using Synthetic Division 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Example 7 Divide (x3– 3x2 – 5x – 25) ÷ (x – 5) using Synthetic Division 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Example 8 Divide (3x4 – x3 + 5x – 1) ÷ (x + 2) using Synthetic Division 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Your Turn Divide (3x4 – 2x2 + 1) ÷ (x + 2) using Synthetic Division 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Theorems Remainder Theorem is where a polynomial f(x) is divided by x – c, and then the remainder is f(c). Factor Theorem uses the remainder and determines whether the factor is part of the polynomial or not. The remainder has to be ZERO for the factor to work. Synthetic Substitution is another method that allows us to evaluate the polynomial function. 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Example 9 Determine 4, –2, and –5 are zeros of the given polynomial f(x) = x3 – x2 – 22x + 40. 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Example 10 Find the remainder when x3 + 2x2 – 5x – 1 is divided by x – 2. Then, use the factor theorem to prove whether the factor is polynomial. 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Example 11 Find the remainder when 21x3 – 46x + 25 is divided by x – 1. Then, use the factor theorem to prove whether the factor is polynomial. 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Your Turn Find the remainder when x4 + 7x3 + 8x2 + 11x + 5 is divided by x + 6. Then, use the factor theorem to prove whether the factor is polynomial. 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Example 12 Use synthetic substitution to evaluate f(x) = 2x4 – 5x3 – 4x + 8 when x = 3. 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Your Turn Use synthetic substitution to evaluate f(x) = –3x3 + x2 – 12x – 5 when x = 2. 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division

5.5 - Long and Synthetic Division Assignment Worksheet 12/8/2018 7:23 AM 5.5 - Long and Synthetic Division