Question

If -1, 1 & 3 are the zeroes of a cubic polynomial, then the correct expression of the polynomial is .

• 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 $x^3-3x^2-x+3$

We know that if a cubic polynomial (whose roots are $\alpha ,\beta and\gamma$) is of the form, $a{x}^3+b{x}^2+cx+d$.

Then, Sum of zeroes, $\alpha +\beta +\gamma =-\frac{b}{a}$
Sum of zeroes taken two at a time, $\alpha \beta +\beta \gamma +\gamma \alpha =\frac{c}{a}$
Product of zeroes, $\alpha \beta \gamma =-\frac{d}{a}$

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

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

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

Therefore b = -3, c = -1, d = 3
Hence, the cubic polynomial will be,
$x^3-3x^2-x+3$