Remainder Theorem.

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Remainder and Factor Theorems

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Presentation transcript:

Remainder Theorem

When a polynomial f(x) is divided by x – a, we can find the remainder as follows: Let Q(x) and R be the quotient and the remainder respectively. By division algorithm, we have Since the degree of R is less than that of x – a, the degree of R is 0. So, R is a constant.  f(x) = (x – a)  Q(x) + R When x = a, the value of the polynomial is f(a) = (a – a)  Q(a) + R = 0  Q(a) + R = R ∴ Remainder = f(a)

In conclusion, we have the following theorem: Theorem 5.1 Remainder theorem When a polynomial f(x) is divided by x – a, the remainder is equal to f(a). When f(x) = x2 + 2x – 1 is divided by x – 1, remainder = f(1) = (1)2 + 2(1) – 1 = 2

In conclusion, we have the following theorem: Theorem 5.1 Remainder theorem When a polynomial f(x) is divided by x – a, the remainder is equal to f(a). When f(x) = x2 + 2x – 1 is divided by x + 2, x + 2 = x – (–2) remainder = f(–2) = (–2)2 + 2(–2) – 1 = –1

Follow-up question Find the remainder when the polynomial x3 – 2x2 – 3x + 5 is divided by x + 1. Let f(x) = x3 – 2x2 – 3x + 5. By the remainder theorem, x + 1 = x – (–1) remainder = f(–1) = (–1)3 – 2(–1)2 – 3(–1) + 5 = –1 – 2 + 3 + 5 = 5

What is the remainder when a polynomial f(x) is divided by mx – n? The remainder theorem can be extended as follows: Theorem 5.2 When a polynomial f(x) is divided by mx – n, the remainder is equal to . | ø ö ç è æ m n f

Find the remainder when 9x2 + 6x – 7 is divided by 3x + 1. Let f(x) = 9x2 + 6x – 7. By the remainder theorem, remainder = | ø ö ç è æ – 3 1 f 3x + 1 = 3x – (–1) 7 3 1 6 9 2 – | ø ö ç è æ + = = 1 – 2 – 7 = –8

Follow-up question When 4x3 + kx2 – x + 2 is divided by 4x – 1, the remainder is 2. Find the value of k. Let f(x) = 4x3 + kx2 – x + 2. By the remainder theorem, 3 k = 16