If 2 is a zero of the polynomial $p (x)$ and one zero is half of the other zero. Which of the following expressions can $p (x)$ be?

p (x) = x^{2} - 6 x + 5

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p (x) = x^{2} - 3 x + 2

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p (x) = x^{2} - 10 x + 16

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p (x) = x^{2} - 6 x + 8

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Solution

The correct options are
B $p (x) = x^{2} - 3 x + 2$
D $p (x) = x^{2} - 6 x + 8$
$∵$
One zero is half the other, the other zero of the polynomial can be either half of 2 or twice of 2.
$\Rightarrow$ Other zero can be 1 or 4.

$∴$ Two polynomials are possible.
They are:

(i) When the two zeroes are 2 and 1
$\Rightarrow$ $p (x) = (x - 1) (x - 2)$
$\Rightarrow$ $p (x) = x^{2} - 3 x + 2$

(i) When the two zeroes are 2 and 4
$\Rightarrow$ $p (x) = (x - 4) (x - 2)$
$\Rightarrow$ $p (x) = x^{2} - 6 x + 8$

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