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This is classified as a challenging question on Factor Remainder Theorem for O-Level Additional Maths.0rchid123 wrote:Please help with the following question
When f(x) is divided by x-1 and x+2 the remainders are 4 and -2 respectively. Hence find the remainder when f(x) is divided by X^2+X-2
By remainder theorem
f(1) = 4
f(-2) = -2
x^2 + x - 2 = (x + 2)(x - 1)
Do you know?
1) f(x) = Q(x) * Divisor + Remainder
2) The degree of the remainder is always less than that of the divisor. Example, a quadratic divisor (degree = 2) is given in this question, the remainder would have a degree of 1 or 0 so in general, we let remainder be Ax+B
f(x) = Q(x) *(x + 2)(x - 1) + (Ax + B)
When x = 1, Thinking process: How do we know what value of x to sub?
f(1) = A + B Notice Q(x) 'disappears'
4 = A + B --- (1)
When x = -2, Thinking process: How do we know what value of x to sub?
f(-2) = -2A + B Notice Q(x) 'disappears'
-2 = -2A + B --- (2)
Values of f(1) and f(-2) are obtained from the first few lines of the solution.
A = 2
B = 2
Hence remainder = 2x + 2 = 2(x + 1)