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secondary 4 | A Maths
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Help me!! I have math EOYs tomorrow
This means it can be smaller than the divisor by 1 or more.
Since the divisor (x - 1)(x - 2) = x² - 3x + 2, the degree of the divisor is 2. So the degree of the remainder can be at most 1 or at lowest 0.
So we can let the remainder be in the form ax + b
f(x) = (x - 1)(x - 2)(quotient) + ax + b
Since the remainder is 2 when divided by (x - 1),
f(1) = 2
(2 - 1)(2 - 2)(quotient) + a(1) + b = 2
a + b = 2
b = 2 - a
Since the remainder is 3 when divided by (x - 2),
f(2) = 3
(1 - 1)(1 - 2)(quotient) + a(2) + b = 3
2a + b = 3
Sub b = 2 - a,
2a + 2 - a = 3
a = 1
So b = 2 - 1 = 1
The remainder is x + 1.
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Anyway , do take note that the degree of the remainder is at most 1 smaller than the degree of the divisor. It may be 1 smaller or more degrees smaller.
So if you get A = 0 for similar questions when your divisor has a degree of 2, don't worry as it can just mean your remainder is a constant