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Question

# If x2âˆ’3x+2 is a factor of f(x)=x4âˆ’px2+q ,then (p,q)=

A
(4,5)
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B
(4,5)
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C
(5,4)
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D
(5,4)
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Solution

## The correct option is D (5,4)Let us first factorize x2−3x+2 as follows:x2−3x+2=x2−x−2x+2=x(x−1)−2(x−1)=(x−1)(x−2)It is given that x2−3x+2 is a factor of the polynomial f(x)=x4−px2+q that is (x−1)(x−2) are the factors of f(x)=x4−px2+q and therefore, x=1 and x=2 are the zeroes of f(x).Now, we substitute x=1 and x=2 in f(x)=x4−px2+q as shown below:f(1)=14−p(1)2+q⇒0=1−p+q⇒p−q=1....(1)f(2)=24−p(2)2+q⇒0=16−4p+q⇒4p−q=16....(2)Subtract equation 1 from equation 2:(4p−p)+(−q+q)=16−1⇒3p=15⇒p=153⇒p=5Substituting the value of p in equation 1, we get:5−q=1⇒q=5−1⇒q=4Hence, (p,q)=(5,4).

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