Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as \(2, –7, –14\) respectively.
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Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as \(2, –7, –14\) respectively.
Answer
Let us consider the cubic polynomial is \(ax^3 + bx^2 + cx + d\) and the values of the zeroes of the polynomials be \(α, β, γ\).
As per the given question,
\(α + β + γ =2\)
\(αβ + βγ + γα = -7\)
\(α β γ = -14\)
Hence, the cubic polynomial is,
\(x^3 -(α + β + γ )x^2 +(αβ + βγ + γα)x -(α β γ)\)
\(=x^3 -(2)x^2 +(-7)x -(-14)\)
\(=x^3 -2x^2 -7x + 14\)
As per the given question,
\(α + β + γ =2\)
\(αβ + βγ + γα = -7\)
\(α β γ = -14\)
Hence, the cubic polynomial is,
\(x^3 -(α + β + γ )x^2 +(αβ + βγ + γα)x -(α β γ)\)
\(=x^3 -(2)x^2 +(-7)x -(-14)\)
\(=x^3 -2x^2 -7x + 14\)
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