What is the Remainder Theorem?
Recall that for long division for integers, the dividing process stops when the remainder is less than the divisor.
dividend = divisor × quotient + remainder
The process is similar for division of polynomials. The dividing stops when the remainder is less that the degree of the divisor.
The Remainder Theorem states that
If a polynomial f(x) is divided by a linear divisor (x – a), the remainder is f(a)
Hence, when the divisor is linear, the remainder can be found by using the Remainder Theorem.How to use the Remainder Theorem?
Find the remainder when 4x3 – 5x + 1 is divided by
a) x – 2
b) x + 3
c) 2x – 1
Let f(x) = 4x3– 5x + 1
a) When f(x) is divided by x – 2, remainder,
R = f(2) = 4(2)3– 5(2) + 1 = 23
b) When f(x) is divided by x + 3, remainder,
R = f(–3) = 4(–3)3– 5(–3) + 1 = –92
c) When f(x) is divided by 2x – 1, remainder,
The expression 4x2 – px + 7 leaves a remainder of –2 when divided by x – 3. Find the value of p.
Let f(x) = 4x2– px + 7
By the Remainder Theorem,
f(3) = –2
4(3)2– 3p + 7 = –2
p = 15
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