Question

Find a cubic polynomial with the sum, sum of the product of its zeros taken two at a time, and the product of its zeros as 3, -1, -3 respectively.

Answer 1

the first picture is the actual answer to your question.
the second picture is the formula used .
the third picture is an example of similar sums.

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Answer 2

The cubic polynomial is $\bf {x}^{3} -3{x}^{2} - x +3 = 0$

Given:

• Sum of zeros is 3,
• Sum of the product of its zeros taken two at a time is -1,
• and the product of its zeros as is -3.

To find:

• Find the cubic polynomial.

Solution:

Concept/Formula to be used:

If $\alpha, \beta , \: and \: \gamma$ are the zeros of cubic polynomial then it is given by

$\bf k[ {x}^{3} - ( \alpha + \beta + \gamma ) {x}^{2} + ( \alpha \beta + \beta \gamma - \alpha \gamma )x + \alpha \beta \gamma ] = 0$

where k is non-zero number.

Step 1:

Write the values given in the question.

$\alpha + \beta + \gamma = 3...eq1$

$\alpha \beta + \beta \gamma + \alpha \gamma = - 1...eq2$

and

$\alpha \beta \gamma = - 3...eq3$

Step 2:

Find the value from eq1,eq2 and eq3 in cubic polynomial.

${x}^{3} - (3) {x}^{2} + ( - 1)x - ( - 3) = 0$

or

${x}^{3} -3{x}^{2} - x + 3 = 0$

Thus,

The cubic polynomial is $\bf {x}^{3} -3{x}^{2} - x +3 = 0$

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