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State whether the following statement is true or false.

The $3, - 1, \frac{- 1}{3}$ are the zeroes of the cubic polynomial $p (x) = 3 x^{3} - 5 x^{2} - 11 x - 3$ and the relationship between the zeroes and the coefficients are verified.

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True

False

answer is A.

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Detailed Solution

Given polynomial,

p (x) = 3 x^{3} - 5 x^{2} - 11 x - 3

. For

x = 3

p (3) = 3 {(3)}^{3} - 5 {(3)}^{2} - 11 \times 3 - 3

\Rightarrow p (3) = 3 \times 27 - 5 \times 9 - 33 - 3

\Rightarrow p (3) = 81 - 45 - 36

\Rightarrow p (3) = 0

For

x = - 1

p (- 1) = 3 {(- 1)}^{3} - 5 {(- 1)}^{2} - 11 \times (- 1) - 3

\Rightarrow p (- 1) = - 3 - 5 + 11 - 3

\Rightarrow p (- 1) = 11 - 11

\Rightarrow p (- 1) = 0

For

x = \frac{- 1}{3}

p (\frac{- 1}{3}) = 3 {(\frac{- 1}{3})}^{3} - 5 {(\frac{- 1}{3})}^{2} - 11 \times (\frac{- 1}{3}) - 3

\Rightarrow p (\frac{- 1}{3}) = 3 (\frac{- 1}{27}) - 5 (\frac{1}{9}) + \frac{11}{3} - 3

\Rightarrow p (\frac{- 1}{3}) = - \frac{1}{9} - \frac{5}{9} + \frac{11}{3} - 3

\Rightarrow p (\frac{- 1}{3}) = \frac{- 1 - 5 + 33 - 27)}{9}

\Rightarrow p (\frac{- 1}{3}) = 0

So,

3, - 1, \frac{- 1}{3}

are the zeroes of the cubic polynomial

p (x) = 3 x^{3} - 5 x^{2} - 11 x - 3

.
For verification,
Let,

α = 3, β = - 1, and γ = \frac{- 1}{3}

If α, β, and γ are zeros of a cubic polynomial

a x^{3} - b x^{2} + cx + d

then,

α + β + γ = \frac{- Coefficient of x^{2}}{Coefficient of x^{3}}

α β + βγ + γα = \frac{Coefficient of x}{Coefficient of x^{3}}

, and

α βγ = \frac{- Constant Term}{Coefficient of x^{3}}

Then,

α + β + γ = 3 - 1 - \frac{1}{3} = \frac{9 - 4}{3} = \frac{5}{3}

\Rightarrow α + β + γ = \frac{- (- 5)}{3} = \frac{- Coefficient of x^{2}}{Coefficient of x^{3}}

And,

α β + βγ + γα = 3 \times (- 1) + (- 1) \times \frac{(- 1)}{3} + \frac{(- 1)}{3} \times 3

\Rightarrow α β + βγ + γα = - 3 + \frac{1}{3} - 1 = \frac{- 11}{3}

\Rightarrow α β + βγ + γα = \frac{- 11}{3} = \frac{Coefficient of x}{Coefficient of x^{3}}

And,

α βγ = 3 \times (- 1) \times (\frac{- 1}{3}) = 1

\Rightarrow α βγ = - \frac{- 1}{1} = \frac{- Constant Term}{Coefficient of x^{3}}

In above expressions the relationship between the zeroes and the coefficients is verified.
Therefore, the given statement is true.
Hence, option (1) is correct.

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