f(x) = x^4 + x^3 + 2x^2 + ax + b
where a and b are constants - Edexcel - A-Level Maths Pure - Question 3 - 2011 - Paper 3
Question 3
f(x) = x^4 + x^3 + 2x^2 + ax + b
where a and b are constants.
When f(x) is divided by (x - 1), the remainder is 7.
(a) Show that a + b = 3.
When f(x) is divided ... show full transcript
Worked Solution & Example Answer:f(x) = x^4 + x^3 + 2x^2 + ax + b
where a and b are constants - Edexcel - A-Level Maths Pure - Question 3 - 2011 - Paper 3
Step 1
(a) Show that a + b = 3.
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Answer
To show that a+b=3, we will evaluate the polynomial at x=1.
Substitute x=1 into f(x): f(1)=14+13+2(1)2+a(1)+b=1+1+2+a+b=4+a+b
Since the polynomial is divided by (x−1), the remainder is given as 7: 4+a+b=7
Rearranging gives: a+b=7−4 a+b=3
Thus, we have shown that a+b=3.
Step 2
(b) Find the value of a and the value of b.
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Answer
To find the values of a and b, we will evaluate the polynomial at x=−2.
Substitute x=−2 into f(x): f(−2)=(−2)4+(−2)3+2(−2)2+a(−2)+b
Calculating this gives: f(−2)=16−8+8−2a+b=16−8+8−2a+b=16−2a+b
Since the polynomial is divided by (x+2), the remainder is given as -8: 16−2a+b=−8
Rearranging gives: −2a+b=−8−16 −2a+b=−24
Now we have the system of equations:
a+b=3
−2a+b=−24
We can eliminate b by rearranging the first equation to find b in terms of a: b=3−a
Substitute into the second equation: −2a+(3−a)=−24 −3a+3=−24 −3a=−24−3 −3a=−27 a=9
