Question

If x - 4 and x-1/4 are factors of px2 + 4. 5x + r, then which of the following is true?
a) p = r/3
b) p = 3r
c) p = r
d) p = r/9

Answer 1

$Let \: f(x) = px^{2} + 4.5x + r$

$i) If \: (x-4) \:is \:a \: factor \:f(x) \then \: f(4)=0$

$\implies p\times 4^{2} + 4.5\times 4 + r = 0$

$\impliea 16p + 18 + r = 0 \: --(1)$

$ii) If \: Big(x-\frac{1}{4}\Big) \:is \:a \: factor \:f(x)\ \then \: f\big(\frac{1}{4}\big)=0$

$\implies p\times \big(\frac{1}{4}\big)^{2} + 4.5\times \frac{1}{4} + r = 0$

$\implies \frac{p}{16} + \frac{4.5}{4}+ r = 0$

/* Multiplying each term by 16, we get */

$\implies p + 4.5 \times 4 + 16r = 0\\\implies p + 18 + 16r = 0 \: --(2)$

$\pink { (1) } = \blue {(2)}$

$\implies 16p + 18 + r = p + 18 + 16r$

$\implies 16p + 18 -p - 18 = 16r - r$

$\implies 15p = 15r$

/* Dividing bothsides by 15, We get */

$\implies p = r$

Therefore.,

$Option \: \pink { ( c ) } \: is \: correct$

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