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 compute the remainders when dividing the polynomial f(x) by (x - 2) and (x + 1).
Remainder when divided by (x - 2):
Using the Remainder Theorem, we find:
f(2)=24+5(2)3+2a+b=16+40+2a+b=56+2a+b
Remainder when divided by (x + 1):
Similarly, we compute:
f(−1)=(−1)4+5(−1)3+a(−1)+b=1−5−a+b=−4−a+b
Setting the two expressions equal to each other:
Since both remainders are equal:
56+2a+b=−4−a+b
Simplifying gives:
56+2a=−4−a3a=−60a=−20
Thus, the value of a is -20.
Step 2
Find the value of b.
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Answer
Given that (x + 3) is a factor of f(x), this implies that f(-3) = 0.
Substituting -3 into f(x):f(−3)=(−3)4+5(−3)3+a(−3)+b=81−135−3a+b
where we substitute a = -20:
=81−135−3(−20)+b=81−135+60+b=6+b
Setting f(-3) to zero:
Since (x + 3) is a factor:
6+b=0b=−6
