1. 5.5 - Long and Synthetic Division
Worksheet Key
12/8/2018 7:23 AM
5.5 - Long and Synthetic Division

2. 5.5 - Long and Synthetic Division
Worksheet Key
12/8/2018 7:23 AM
5.5 - Long and Synthetic Division

3. 5.5 - Long and Synthetic Division
Worksheet Key
12/8/2018 7:23 AM
5.5 - Long and Synthetic Division

4. 5.5 - Long and Synthetic Division
Worksheet Key
12/8/2018 7:23 AM
5.5 - Long and Synthetic Division

5. Revised ©2013, vdang@houstonisd.org
Polynomial Division
Revised ©2013,
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5.5 - Long and Synthetic Division

6. 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.
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5.5 - Long and Synthetic Division

7. 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
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5.5 - Long and Synthetic Division

8. 5.5 - Long and Synthetic Division
Example 1
Divide x3 – 5x2 – 12x + 36 ÷ x – 2 using Long Division
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5.5 - Long and Synthetic Division

9. 5.5 - Long and Synthetic Division
Example 2
Divide (x3 – 28x – 48) ÷ (x + 4) using Long Division
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5.5 - Long and Synthetic Division

10. 5.5 - Long and Synthetic Division
Your Turn
Divide (6x3 + 5x2 + 9) ÷ (2x + 3) using Long Division
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5.5 - Long and Synthetic Division

11. 5.5 - Long and Synthetic Division
Example 3
Divide (2x2 + 3x – 4) ÷ (x – 2) using Long Division
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5.5 - Long and Synthetic Division

12. 5.5 - Long and Synthetic Division
Example 4
Divide (2x4 + 4x3 – 5x2 + 3x – 2) ÷ (x2 + 2x – 3) using Long Division
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5.5 - Long and Synthetic Division

13. 5.5 - Long and Synthetic Division
Example 5
Divide (12x4 – 5x2 – 3) ÷ (3x2 – 5) using Long Division
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5.5 - Long and Synthetic Division

14. 5.5 - Long and Synthetic Division
Your Turn
Divide (x4 + 3x2 + 1) ÷ (x2 – 2x + 3) using Long Division
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5.5 - Long and Synthetic Division

15. 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
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5.5 - Long and Synthetic Division

16. 5.5 - Long and Synthetic Division
Example 6
Divide (x3 – 5x2 – 12x + 36) ÷ (x – 2) using Synthetic Division
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5.5 - Long and Synthetic Division

17. 5.5 - Long and Synthetic Division
Example 6
Divide (x3 – 5x2 – 12x + 36) ÷ (x – 2) using Synthetic Division
2 · 1 = 2
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5.5 - Long and Synthetic Division

18. 5.5 - Long and Synthetic Division
Example 6
Divide (x3 – 5x2 – 12x + 36) ÷ (x – 2) using Synthetic Division
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5.5 - Long and Synthetic Division

19. 5.5 - Long and Synthetic Division
Example 7
Divide (x3– 3x2 – 5x – 25) ÷ (x – 5) using Synthetic Division
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5.5 - Long and Synthetic Division

20. 5.5 - Long and Synthetic Division
Example 8
Divide (3x4 – x3 + 5x – 1) ÷ (x + 2) using Synthetic Division
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5.5 - Long and Synthetic Division

21. 5.5 - Long and Synthetic Division
Your Turn
Divide (3x4 – 2x2 + 1) ÷ (x + 2) using Synthetic Division
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5.5 - Long and Synthetic Division

22. 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.
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5.5 - Long and Synthetic Division

23. 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.
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5.5 - Long and Synthetic Division

24. 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.
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5.5 - Long and Synthetic Division

25. 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.
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5.5 - Long and Synthetic Division

26. 5.5 - Long and Synthetic Division
Your Turn
Find the remainder when x4 + 7x3 + 8x2 + 11x + 5 is divided by x Then, use the factor theorem to prove whether the factor is polynomial.
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5.5 - Long and Synthetic Division

27. 5.5 - Long and Synthetic Division
Example 12
Use synthetic substitution to evaluate f(x) = 2x4 – 5x3 – 4x + 8 when x = 3.
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5.5 - Long and Synthetic Division

28. 5.5 - Long and Synthetic Division
Your Turn
Use synthetic substitution to evaluate f(x) = –3x3 + x2 – 12x – 5 when x = 2.
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5.5 - Long and Synthetic Division

29. 5.5 - Long and Synthetic Division
Assignment
Worksheet
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5.5 - Long and Synthetic Division