2.3 Remainder Theorem 2.3 Factor Theorem 2.3 Rational Zeros

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Pre-Calculus Honors

Chapter 2: Polynomial & Rational Functions

Monday October 4

2.3 Synthetic Division

Tuesday October 5

2.3 Remainder Theorem

Wednesday October 6

2.3 Factor Theorem

Thursday October 7

2.3 Rational Zeros

Friday October 8

2.3 Rational Root Theorem

2.3 Real Zeros of Polynomial Functions: Synthetic Division Synthetic Division

When dividing polynomials, there is a shortcut method that you can use called synthetic division. To divide 3 + 2 + + by - , you will follow the pattern as shown below. The vertical pattern is to add terms; the diagonal pattern is to multiply by "k" as you will see below. Example 1: Solve with synthetic division. (4 - 102 - 2 + 4) ? ( + 3)

Example 2: Solve with synthetic division. (53 + 82 - + 6) ? ( - 2)

Example 3: Solve with synthetic division. (23 + 13 - 10) ? ( + 4)

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Pre-Calculus Honors

Name ___________________________________

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2.3: Synthetic Division

Divide. Use synthetic division. Show all work neatly on another sheet of paper.

1) (n3 - 10n2 + 19n - 8) ? (n - 1)

2) (9n3 + 15n2 + 9n + 4) ? (n + 1)

3) (x3 + 4x2 + 1) ? (x + 4)

4) (x3 + 9x2 - 5x - 7) ? (x - 1)

5) (k3 + 6k2 - 12k + 12) ? (k - 1)

6) (m3 + 3m2 + 3m) ? (m + 3)

7) (x4 + 18x3 + 85x2 + 48x + 74) ? (x + 8)

8) (r4 + 10r3 - r2 - 100r - 82) ? (r + 9)

9) (r4 - 11r3 + 34r2 - 34r + 36) ? (r - 4)

10) (m4 - m3 - 92m2 + 21m - 13) ? (m - 10)

11) (19r3 + 4r4 - 73r2 + 41 - 64r) ? (7 + r)

12) (x4 + 2x3 - 9x - 9) ? (x + 2)

13) ( p5 + 5 p4 - 26 p3 - 3 p2 + 26 p + 5) ? ( p - 3)

14) (x5 - x4 - 97x3 - 53x2 + 96x + 62) ? (x + 9)

15) (n5 - 2n3 - 21n2 + 7n - 15) ? (n - 3)

16) (v5 - 7v4 + 21v3 - 34v2 - v - 33) ? (v - 4)

17) (x5 - 3x4 - 31x3 - 64x2 - 72x + 55) ? (x - 8)

18) (r5 - 12r4 + 30r3 - 8r2 - r - 9) ? (r - 3)

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2.3 Real Zeros of Polynomial Functions: Remainder Theorem Remainder Theorem

The Remainder Theorem says, "If a polynomial () is divided by ( - ), then the remainder is (). Example 1: Use the Remainder Theorem to evaluate () = 33 + 82 + 5 - 7 when = -2.

Example 2: Use the Remainder Theorem to find each function value given: () = 43 + 102 - 3 - 8

Find: (-1), (4), (1) , (-3)

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Pre-Calculus Honors

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2.3: Remainder Theorem

Evaluate each function using Remainder Theorem. Show all work neatly on another sheet of paper.

1) f (m) = m4 + m3 - 13m2 - m + 12 at m = -4

2) f (x) = x3 + 2x2 + 4x + 1 at x = -2

3) f (x) = x2 - 7x + 11 at x = 5

4) f (a) = -2a3 + 7a2 + 9a + 21 at a = 5

5) f (n) = n3 + 10n2 + 30n + 19 at n = -4

6) f (n) = n4 - 42n2 - 42n - 46 at n = -6

7) f (x) = x3 + 6x2 + 4x - 18 at x = -4

8) f (a) = a4 + a3 - 30a2 + 5a + 38 at a = -6

9) f (n) = 6n5 - 32n4 + 13n3 - 12n2 - 20n + 26

at n = 5

10) f (n) = n6 + 5n5 + 12n4 + 24n3 + 15n2 - 5n + 7

at n = -3

11) f (x) = x3 - 35x - 1 at x = -6

12) f (n) = n4 - 8n3 + 13n2 + 14n - 15 at n = 4

13) f (m) = m5 - 8m4 + 9m3 + 17m2 + 36 at m = 6

14) f (n) = n3 + n2 - 18n - 6 at n = 4

15) f (a) = a2 - 33 at a = -5

16) f (x) = x6 - 10x5 + 18x4 + 20x3 + 12x2 + 13x + 11 at x = 4

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