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Is the following statement true or false?

$(x - 3)$ is a factor of the polynomial $p (x) = 2 x^{3} + 7 x^{2} - 24 x - 45.$

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False

True

answer is A.

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Detailed Solution

The given polynomial is

p (x) = 2 x^{3} + 7 x^{2} - 24 x - 45.

We know by Factor theorem that if

(x - k)

is a factor of the given polynomial

p (x),

then

p (k) = 0.

Since, to check whether

(x - 3)

is a factor of the polynomial

p (x) = 2 x^{3} + 7 x^{2} - 24 x - 45,

we check whether

p (3) = 0.

Substitute the value of

x

p (x),

we get,

\begin{array}{l} p (3) = [2 \times (^{3) 3}] + [7 \times (^{3) 2}] - (24 \times 3) - 45 \\ \Rightarrow p (3) = 54 + 63 - 72 - 45 \\ \Rightarrow p (3) = 117 - 117 \\ \Rightarrow p (3) = 0 \end{array}

Since

p (3) = 0,

therefore

(x - 3)

is a factor of

p (x) = 2 x^{3} + 7 x^{2} - 24 x - 45.

Hence, the given statement is true.
Thus option (1) is correct.

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