Question

$\text{If -1,1 \& 3 are the zeroes of a cubic}\text{polynomial, which of the following is the}\text{correct expression of the polynomial?}$

• A. $x^3+3x^2-x+3$
• B. $x^3-3x^2+x+3$
• C. $x^3-3x^2-x-3$
• D. $x^3-3x^2-x+3$

Answer 1

The correct option is D $x^3-3x^2-x+3$

$\text{We know that if a cubic polynomial is}a{x}^3+b{x}^2+cx+d,\text{where the zeros}\text{are}\alpha ,\beta \text{and}\gamma \text{then,}\text{Sum of zeroes}=\alpha +\beta +\gamma =-\frac{b}{a}\text{Sum of zeroes taken two at a time}=\alpha \beta +\beta \gamma +\gamma \alpha =\frac{c}{a}\text{Product of zeroes}=\alpha \beta \gamma =\frac{-d}{a}$

$\text{Assume a = 1,}\frac{-b}{a}=\frac{-b}{1}=-1+1+3=3\Rightarrow b=-3$

$\frac{c}{a}=\frac{c}{1}=(-1)\left(1\right)+\left(1\right)\left(3\right)+\left(3\right)(-1)=-1+3-3=-1\Rightarrow c=-1$

$\frac{-d}{a}=\frac{-d}{1}=(-1)\left(1\right)\left(3\right)=-3\Rightarrow d=3$

$\therefore a=1,b=-3,c=-1,d=3$

$\text{Hence, the cubic polynomial will be}{x}^3-3x^2-x+3.$