If the zeroes of the polynomials $p (x) = 2 x - 4$ and $q (x) = 5 x - 15$ are multipled, we get a zero of the polynomial $r (x) = a x^{2} + b x + 10$ . Which of the following relations is true?

a + 6 b + 10 = 0

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6 a + 6 b - 10 = 0

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36 a + 6 b - 10 = 0

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36 a + 6 b + 10 = 0

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Solution

The correct option is D $36 a + 6 b + 10 = 0$
Zero of the polynomial $p (x) = 2 x - 4$
$\Rightarrow$ $2 x - 4 = 0$
$\Rightarrow$ $2 x = 4$
$\Rightarrow$ $x = 2$

Similarly,
Zero of the polynomial $q (x) = 5 x - 15$
$\Rightarrow$ $5 x - 15 = 0$
$\Rightarrow$ $5 x = 15$
$\Rightarrow$ $x = 3$

$∴$ Multiplying these zeroes ,we get $2 \times 3 = 6$

Given, 6 is a zero of the polynomial $r (x) = a x^{2} + b x + 10$
$\Rightarrow$ $r (6) = 0$
$\Rightarrow$ $a (6^{2}) + b (6) + 10 = 0$
$\Rightarrow$ $36 a + 6 b + 10 = 0$

$∴$ $36 a + 6 b + 10 = 0$ is the required answer.

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