1. Polynomial and Synthetic Division
Section 2.3
Precalculus PreAP/Dual, Revised ©2017
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§2.3: Polynomial and Synthetic Division

2. 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.
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§2.3: Polynomial and Synthetic Division

3. §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
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§2.3: Polynomial and Synthetic Division

4. §2.3: Polynomial and Synthetic Division
Example 1
Divide 𝒙 𝟑 −𝟓 𝒙 𝟐 −𝟏𝟐𝒙+𝟑𝟔 ÷ 𝒙−𝟐 using Long Division
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§2.3: Polynomial and Synthetic Division

5. §2.3: Polynomial and Synthetic Division
Example 2
Divide 𝒙 𝟑 −𝟐𝟖𝒙−𝟒𝟖 ÷ 𝒙+𝟒 using Long Division
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§2.3: Polynomial and Synthetic Division

6. §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
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§2.3: Polynomial and Synthetic Division

7. §2.3: Polynomial and Synthetic Division
Your Turn
Divide 𝟔 𝒙 𝟑 +𝟓 𝒙 𝟐 +𝟗 ÷ 𝟐𝒙+𝟑 using Long Division
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§2.3: Polynomial and Synthetic Division

8. §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
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§2.3: Polynomial and Synthetic Division

9. §2.3: Polynomial and Synthetic Division
Example 4
Divide 𝒙 𝟑 −𝟓 𝒙 𝟐 −𝟏𝟐𝒙+𝟑𝟔 ÷ 𝒙−𝟐 using Synthetic Division
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§2.3: Polynomial and Synthetic Division

10. §2.3: Polynomial and Synthetic Division
Example 4
Divide 𝒙 𝟑 −𝟓 𝒙 𝟐 −𝟏𝟐𝒙+𝟑𝟔 ÷ 𝒙−𝟐 using Synthetic Division
2 · 1 = 2
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§2.3: Polynomial and Synthetic Division

11. §2.3: Polynomial and Synthetic Division
Example 4
Divide 𝒙 𝟑 −𝟓 𝒙 𝟐 −𝟏𝟐𝒙+𝟑𝟔 ÷ 𝒙−𝟐 using Synthetic Division
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§2.3: Polynomial and Synthetic Division

12. §2.3: Polynomial and Synthetic Division
Your Turn
Divide 𝟑𝒙 𝟒 − 𝒙 𝟑 +𝟓𝒙−𝟏 ÷ 𝒙+𝟐 using Synthetic Division
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§2.3: Polynomial and Synthetic Division

13. 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 .
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§2.3: Polynomial and Synthetic Division

14. §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.
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§2.3: Polynomial and Synthetic Division

15. §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 .
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§2.3: Polynomial and Synthetic Division

16. §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.
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§2.3: Polynomial and Synthetic Division

17. §2.3: Polynomial and Synthetic Division
Assignment
Page , 15, 17, 21, 25, 27, 31, 33, 35, 39, 43, 47, 51, 59, 61
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§2.3: Polynomial and Synthetic Division