Oxford University MAT 2017 Solved Papers

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Question : 8 of 10

Marks: +1, -0

In this question

a

and

b

are real numbers, and

a

is non-zero.
When the polynomial

x^2-2 a x+a^4

is divided by

x+b

the remainder is 1 .
The polynomial

b x^2+x+1

has

a x-1

as a factor.
It follows that

b

equals

1 only
0 or $-2$
1 or 2
1 or 3
$-1$ or 2 .

Solution:

By the remainder theorem

f(-b)=1

, so

b^2+2 a b+a^4=1 .

By the factor theorem

f(\;{1}/{a})=0

,

\;{b}/{a^2}+\;{1}/{a}+1 \;\; =0

b+a+a^2 \;\; =0

-a-a^2 \;\; =b .

Combining these two equations we find that

2 a^4-a^2=1

and so

a=-1

or 1 . Hence

b=-2

or 0 , and so the answer is (b).

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