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= remainder = P (- 2) . Polynomials: The Remainder and Factor Theorems. The remainder theorem states that if a polynomial, P ( x ), is divided by x – c, then the remainder equals P ( c ).

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= remainder = P(- 2)

Polynomials: The Remainder and Factor Theorems

The remainder theorem states that if a polynomial, P(x), is divided by x – c, then the remainder equals P(c).

Example 1: For the polynomial, P(x) = 2x3 – 8x2 + 45, (a) find P(- 2) by direct evaluation, (b) find P(- 2) using the remainder theorem.

(a) P(- 2) = 2(- 2)3 – 8(- 2)2 + 45

= - 3

(b) Here c = - 2, so divide (synthetically) P(x) by x + 2.

2 - 8 0 45

- 4 24 - 48

- 2 | 2 - 12 24 - 3

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Polynomials: The Remainder and Factor Theorems

The factor theorem states that for a polynomial, P(x), if x – c is a factor, then P(c) = 0. Also, if P(c) = 0, then x – c is a factor.

Example 2: For the polynomial, P(x) = 2x3 – x2 + 3x – 4,

use the factor theorem to show that x – 1 is a

factor.

P(1) = 2(1)3 – (1)2 + 3(1) – 4

= 0.

Since P(1) = 0, x– 1 is a factor.

Slide 2

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Polynomials: The Remainder and Factor Theorems

Try: For the polynomial, P(x) = x3 – x2 + x – 6,

(a) find P(5) using the remainder theorem, (b) use the factor theorem to show that x – 2 is a factor.

(a) 1 - 1 1 - 6

5 20 105

5 | 1 4 21 99

= P(5)

(b) P(2) = (2)3 – (2)2 + (2) – 6 = 0

Since P(2) = 0, x– 2 is a factor.

Slide 3

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Polynomials: The Remainder and Factor Theorems

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