Question

Find a cubic polynomial with the sum, sum of products of its zeroes taken two at a time, and the product of its zeroes as 2, -7, -14 respectively.

Answer 1

Hint: To solve this problem, one should have theoretical knowledge of cubic equations and the relation between their coefficients and their roots. The formulas that will be used to verify this relationship are \[-\alpha +\beta +\gamma =\dfrac{-b}{a},\alpha \beta +\gamma \beta +\gamma \alpha =\dfrac{c}{a}\] and \[\alpha \beta \gamma =\dfrac{-d}{a}\]. So, using the information given in the question and these three formulas, we will first reform the cubic equation and then we will obtain the equation of the cubic polynomial.

Complete step by step answer:
Let the equation be \[a{{x}^{3}}+b{{x}^{2}}+cx+d=0\]. We will divide both the sides by a in the above equation, and then apply the formula for the relationship between the roots and the coefficients, which is given by \[-\alpha +\beta +\gamma =\dfrac{-b}{a},\alpha \beta +\gamma \beta +\gamma \alpha =\dfrac{c}{a}\] and \[\alpha \beta \gamma =\dfrac{-d}{a}\], as it is given that sum, sum of product and the product of its zeroes are 2, -7, -14 respectively.
So, we will write them as,
\[-\alpha +\beta +\gamma =2,\alpha \beta +\gamma \beta +\gamma \alpha =-7\] and \[\alpha \beta \gamma =-14\]
So, here we can see that we can substitute these formulas in the equation.
\[{{x}^{3}}+\dfrac{b}{a}{{x}^{2}}+\dfrac{c}{a}x+\dfrac{d}{a}=0\]
\[{{x}^{3}}+\left( \alpha +\beta +\gamma \right){{x}^{2}}+\left( \alpha \beta +\beta \gamma +\alpha \gamma \right)x-\alpha \beta \gamma =0\]
We put the given values to get,
\[\Rightarrow {{x}^{3}}-\left( 2 \right){{x}^{2}}+\left( -7 \right)x-\left( -17 \right)=0\]
On simplifying we get,
\[\Rightarrow {{x}^{3}}-2{{x}^{2}}-7x+14=0\]
So, this is the required cubic polynomial.

Note: Always remember that general equation of cubic polynomial is \[a{{x}^{3}}+b{{x}^{2}}+cx+d=0\], where a, b, c and d belongs to set of real numbers and also keep the logic remember that number of roots of polynomial is equals to the highest power of x in polynomial. The most common mistake here is that the sign of sum and products are forgotten at times. So, try to remember them.