Question

A cubic polynomial whose zeros are1,3,-2

Answer 1

ANSWER:

• CUBIC POLYNOMIAL:
• $p(x) \: = x {}^{3} - 2x {}^{2} - 5x + 6$

GIVEN:

$\alpha \: = 1 \\ \beta \: = 3 \\ \gamma \: = ( - 2)$

TO FIND:

• A cubic polynomial having zeros 1, 3, (-2)

SOLUTION:

• Standard form of cubic polynomial with zeros:

$\implies \: x {}^{3} - ( \alpha + \beta + \gamma )x {}^{2} + ( \alpha \beta + \beta \gamma \: + \gamma \alpha)x - \alpha \beta \gamma$

Now finding the values:

$\implies \: \alpha \: + \beta \: + \gamma \: = 1 + 3 - 2 \\ \implies \: \alpha \: + \beta \: + \gamma \: = 2 \\ \\ \implies \: \alpha \beta+ \beta \gamma+ \gamma \alpha = (1 \times 3) + (3 \times ( - 2)) + ( - 2) \times 1 \\ \implies \: \alpha \beta \: + \beta \gamma + \gamma \alpha = 3 - 6 - 2 \\ \implies \: \alpha \beta + \beta \gamma + \gamma \alpha = \: - 5 \\ \\ \implies \: \alpha \: \beta \: \gamma \: = 1 \times 3 \times ( - 2) \\ \implies \: \alpha \: \beta \: \gamma \: = - 6$

Putting the values in the standard form of cubic polynomial.

$p(x) \: = x {}^{3} - 2x {}^{2} - 5x + 6$

NOTE:

Some important formulas:

• Standard form of cubic polynomial

$\implies \: x {}^{3} - ( \alpha + \beta + \gamma )x {}^{2} + ( \alpha \beta + \beta \gamma \: + \gamma \alpha)x - \alpha \beta \gamma$

• Standard form of quadratic polynomial

$\implies \: x {}^{2} - ( \alpha \: + \beta)x + \alpha \beta$