Polynomial and Synthetic Division Section 2.3 Precalculus PreAP/Dual, Revised ©2017 viet.dang@humbleisd.net 12/8/2018 9:11 AM §2.3: Polynomial 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 cannot be done Leftover is remainder NOTE: Long division can be used no matter what size the divisor is. 12/8/2018 9:11 AM §2.3: Polynomial and Synthetic Division

§2.3: Polynomial and Synthetic Division Elementary Review Solve 𝟐𝟓÷𝟑 through long division Quotient Divisor Dividend 𝟑×𝟖=𝟐𝟒 Subtract from the top number Remainder 12/8/2018 9:11 AM §2.3: Polynomial and Synthetic Division

§2.3: Polynomial and Synthetic Division Example 1 Divide 𝒙 𝟑 −𝟓 𝒙 𝟐 −𝟏𝟐𝒙+𝟑𝟔 ÷ 𝒙−𝟐 using Long Division 12/8/2018 9:11 AM §2.3: Polynomial and Synthetic Division

§2.3: Polynomial and Synthetic Division Example 2 Divide 𝒙 𝟑 −𝟐𝟖𝒙−𝟒𝟖 ÷ 𝒙+𝟒 using Long Division 12/8/2018 9:11 AM §2.3: Polynomial and Synthetic Division

§2.3: Polynomial and Synthetic Division Example 3 Divide 𝟐𝒙 𝟒 +𝟒 𝒙 𝟑 −𝟓 𝒙 𝟐 +𝟑𝒙−𝟐 ÷ 𝒙 𝟐 +𝟐𝒙−𝟑 using Long Division 12/8/2018 9:11 AM §2.3: Polynomial and Synthetic Division

§2.3: Polynomial and Synthetic Division Your Turn Divide 𝟔 𝒙 𝟑 +𝟓 𝒙 𝟐 +𝟗 ÷ 𝟐𝒙+𝟑 using Long Division 12/8/2018 9:11 AM §2.3: Polynomial and Synthetic Division

§2.3: Polynomial 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 9:11 AM §2.3: Polynomial and Synthetic Division

§2.3: Polynomial and Synthetic Division Example 4 Divide 𝒙 𝟑 −𝟓 𝒙 𝟐 −𝟏𝟐𝒙+𝟑𝟔 ÷ 𝒙−𝟐 using Synthetic Division 12/8/2018 9:11 AM §2.3: Polynomial and Synthetic Division

§2.3: Polynomial and Synthetic Division Example 4 Divide 𝒙 𝟑 −𝟓 𝒙 𝟐 −𝟏𝟐𝒙+𝟑𝟔 ÷ 𝒙−𝟐 using Synthetic Division 2 · 1 = 2 12/8/2018 9:11 AM §2.3: Polynomial and Synthetic Division

§2.3: Polynomial and Synthetic Division Example 4 Divide 𝒙 𝟑 −𝟓 𝒙 𝟐 −𝟏𝟐𝒙+𝟑𝟔 ÷ 𝒙−𝟐 using Synthetic Division 12/8/2018 9:11 AM §2.3: Polynomial and Synthetic Division

§2.3: Polynomial and Synthetic Division Your Turn Divide 𝟑𝒙 𝟒 − 𝒙 𝟑 +𝟓𝒙−𝟏 ÷ 𝒙+𝟐 using Synthetic Division 12/8/2018 9:11 AM §2.3: Polynomial and Synthetic Division

Remainder & Factor Theorem Factor Theorem uses the remainder and determine whether the factor is part of the polynomial or not. The remainder has to be ZERO for the factor to work. Remainder Theorem is where a polynomial 𝒇 𝒙 is divided by 𝒙−𝒄, then the remainder is 𝒇 𝒄 . 12/8/2018 9:11 AM §2.3: Polynomial and Synthetic Division

§2.3: Polynomial and Synthetic Division Example 5 Determine whether (𝒙−𝟒), (𝒙+𝟐), and 𝒙+𝟓 are zeros of the given polynomial 𝒇 𝒙 = 𝒙 𝟑 − 𝒙 𝟐 −𝟐𝟐𝒙+𝟒𝟎. 12/8/2018 9:11 AM §2.3: Polynomial and Synthetic Division

§2.3: Polynomial and Synthetic Division Example 6 Use the Remainder Theorem when 𝒇 𝒙 =𝟑 𝒙 𝟑 +𝟖 𝒙 𝟐 +𝟓𝒙−𝟕 when 𝒙=−𝟐 using the form, 𝒇 𝒙 = 𝒙−𝒌 𝒒 𝒙 +𝒓 . 12/8/2018 9:11 AM §2.3: Polynomial and Synthetic Division

§2.3: Polynomial and Synthetic Division Your Turn Use the Remainder Theorem when 𝒇 𝒙 = 𝒙 𝟒 +𝟕 𝒙 𝟑 +𝟖 𝒙 𝟐 +𝟏𝟏𝒙+𝟓 when 𝒙=−𝟔. 12/8/2018 9:11 AM §2.3: Polynomial and Synthetic Division

§2.3: Polynomial and Synthetic Division Assignment Page 144 11, 15, 17, 21, 25, 27, 31, 33, 35, 39, 43, 47, 51, 59, 61 12/8/2018 9:11 AM §2.3: Polynomial and Synthetic Division