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Question

If x3+px2+qx+6 has x2 as a factor and leaves remainder 3 when divided by x3 find the value of p and q

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Solution

If the given polynomial is f(x)=x3+px2+qx+6 then solution will follows as:

When f(x) is divided by x3 and x2, the remainders are 3 and 0 respectively.

f(3)=3 and f(2)=0

(3)3+p(3)2+q(3)+6=3and(2)3+p(2)2+q(2)+6=0

27+9p+3q+6=3 and 8+4p+2q+6=0

9p+3q+33=3 and 4p+2q+14=0

9p+3q=30 and 4p+2q=14

3p+q=10 ... (1) and 2p+q=7 .... (2)

On subtracting (1) and (2), we get

3p+q(2p+q)=10(7)

p=10+7=3

On putting p in (2), we get

2(3)+q=7

q=7+6=1

p=3 and q=1


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