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  • If both x – 2 and x-\frac{1}{2} are factors of px^{2} + 5x + r, show that p = r.

If both x – 2 and x-\frac{1}{2}  are factors of px2 + 5x + r, show that p = r.

Answers (1)

Let f(x)=px^{2}+5x+r

We know that if (x+a)  is factor of the polynomial f(x), then it always satisfies f(-a)=0

(x-2)  is a factor of f(x) then f(2) = 0

So,

p(2)^{2}+5(2)+r=0\\ 4p+10+r=0 \cdots \cdots (1)         …..(1)

Also x-\frac{1}{2}  is a factor of f(x) then f\left (\frac{1}{2} \right )=0

p\left ( \frac{1}{2} \right )^{2}+5\left ( \frac{1}{2} \right )+r=0\\ \frac{p}{4}+\frac{5}{2}+r=0\\ p+10+4r=0 \cdots (2)

Since (x – 2) and x-\frac{1}{2}  are factors of f(x) then equation 1 and 2 are equal

4p+10+r=9+10+4r\\ 4p-p=4r-r\\ 3p=3r\\ p=r

Hence proved.

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infoexpert24

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