f(x) = x^4 + 5x^3 + ax + b,
where a and b are constants - Edexcel - A-Level Maths Pure - Question 8 - 2009 - Paper 2
Question 8
f(x) = x^4 + 5x^3 + ax + b,
where a and b are constants.
The remainder when f(x) is divided by (x - 2) is equal to the remainder when f(x) is divided by (x + 1).
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Worked Solution & Example Answer:f(x) = x^4 + 5x^3 + ax + b,
where a and b are constants - Edexcel - A-Level Maths Pure - Question 8 - 2009 - Paper 2
Step 1
Find the value of a.
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Answer
To find the value of a, we need to evaluate the polynomial at the roots of the divisors.
Evaluate f(2):
f(2) = 2^4 + 5(2^3) + 2a + b = 16 + 40 + 2a + b = 56 + 2a + b.
Evaluate f(-1):
f(-1) = (-1)^4 + 5(-1)^3 + a(-1) + b = 1 - 5 - a + b = -4 - a + b.
Set the two remainders equal:
56 + 2a + b = -4 - a + b.
Simplifying this equation:
56 + 2a = -4 - a
3a = -60
a = -20.
Step 2
Find the value of b.
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Answer
Since (x + 3) is a factor of f(x), we have:
Evaluate f(-3):
f(-3) = (-3)^4 + 5(-3)^3 + a(-3) + b = 81 - 135 - 60 + b.
