Download 2 - Notes - Synthetic Division and The Remainder

2 ­ Notes ­ Synthetic Division and The Remainder TheoremLB SP17
H Alg 2 ­ Unit 4B
Warm­up: Divide using Long Division
Vocabulary
1. (2x + x ­ 15) ÷ (2x ­ 5)
2. (4x2 ­ 12x) ÷ (2x + 3)
Synthetic Division: a shortcut method for dividing a polynomial by a linear factor of the form (x – a). It can be used in place of the standard long division algorithm. 3. (8x5 ­ 3x4 + x2 ­ 5x + 4) ÷ (x + 1)
Synthetic Substitution: a method of evaluating polynomials
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Synthetic Division and What am I learning today?
the Remainder Theorem
How to use synthetic division to evaluate a function and how to determine if (x ­ a) is a factor of P(x)
MGSE9­12.A.APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so How will I show that I learned it?
Apply synthetic division to evaluate a function and decide if the divisor is a factor
p(a) = 0 if and only if (x – a) is a factor of p(x). Synthetic Division
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2 ­ Notes ­ Synthetic Division and The Remainder TheoremLB SP17
2.
(3x3 ­ 2x2 ­ 15x + 14) ÷ (x ­ 2)
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If you have a coefficient other than 1 in your divisor, you need to divide it out of your answer.
H Alg 2 ­ Unit 4B
3. (5x3 + 18x2 ­ 5x + 12) ÷ (x + 4)
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7.
(2x3 ­ 11x2 + 16x ­ 9) ÷ (2x ­ 3)
6. (4x3 ­ 12x2 ­ x + 3) ÷ (2x + 1)
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2 ­ Notes ­ Synthetic Division and The Remainder TheoremLB SP17
H Alg 2 ­ Unit 4B
Let's Practice
The Remainder Theorem
a. (x2 ­ 7x + 12) ÷ (x ­ 3)
If P(x) is divided by (x ­ a), then the remainder is the number P(a). If the remainder is 0, then (x ­ a) is a factor of P(x).
b (4x2 + 27x + 20) ÷ (x + 6)
c. (x3 ­ 8) ÷ (x ­ 2)
d. (25x2 ­ 20x + 4) ÷ (5x ­ 2) Synthetic Substitution uses the same process as synthetic division but is used evaluate P(a). In this case, the "remainder", when a is the input value, represents P(a).
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Use Synthetic Substitution to evaluate P(a), determine if (x ­ a) is a factor, and verify with a 2nd method.
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10. Determine P(2) for P(x) = 3x3 ­ x2 + 8x + 12.
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2 ­ Notes ­ Synthetic Division and The Remainder TheoremLB SP17
H Alg 2 ­ Unit 4B
Let's Practice
Evaluate P(x) = 6x3 + x2 + 19x + 14 using the remainder theorem.
11. Find P(1.5) for P(x) = 2x3 ­ 11x2 + 16x ­ 6.
a. P(1)
4. P(1/2)
b. P(­2)
5. P(5)
c. P(­2/3)
Could you find any factors of P(x)?
Homework
p. 102­104 # 19­29, 50, 57­60
p. 109­110 # 17­19, 40­44
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