Question

If x = -1/3 is a zero of the polynomial p(x) = 27x³ - ax² - x + 3, then find the value of a.

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Answer 1

Question :-

■ If x = -1/3 is a zero of the polynomial p(x) = 27x³ - ax² - x + 3, then find the value of a.

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Solution :-

Given Information :-

• ■ x = $\sf \frac{ - 1}{3}$ is a zero of the polynomial.

• ■ Polynomial ➱ p(x) = 27x³ - ax² - x + 3

To Find :-

• ■ The value of a

Calculation :-

Substitute x = $\sf \frac{ - 1}{3}$ to find value of a

⇒ p(x) = 27x³-ax²-x+3

⇒ p(x) =(3x)³-ax²-x+3

⇒ p($\sf \frac{ - 1}{3}$)

$\sf⇒p( \frac{ - 1}{3} ) = { \{3( \frac{ - 1}{3} ) \}}^{3} - a {( \frac{ - 1}{3}) }^{2} - ( \frac{ - 1}{3} ) + 3 = 0$

⇒ (-1)³-a($\sf \frac{ - 1}{9}$)+$\sf \frac{ - 1}{3}$+3 = 0

⇒ -1 $\sf \frac{ - a}{9}$+$\sf \frac{ 1}{3}$+3 = 0

⇒ $\sf \frac{ - a}{9}$+$\sf \frac{ 1}{3}$+2 = 0

⇒ $\sf \frac{ -a + 3 + 18}{2} = 0$

⇒ -a+3+18 = 0

⇒ a = 3+18

⇒ a = 21

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Final Answer :-

• ■ The value of a is 21.

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Note :-

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Answer 2

Answer: a = 21.

Explanation:

Given:

p(x) = 27x³ - ax² - x + 3

And α = - ⅓.

We know, in a polynomial f(x) has a zero β, then f(β) = 0.

So,

p(α) = 27α³ - aα² - α + 3 = 0

=> 27(- 1/3)³ - a(- 1/3)² - (- 1/3) + 3 = 0

=> - 1 - a/9 + 1/3 + 3 = 0

=> - a/9 + 1/3 + 2 = 0

=> - a/9 = - (7/3)

=> a = 7/3 × 9 = 21.

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