Question

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients: $3x^2-x-4$.

Answer 1

Finding the zeroes of the polynomial:

Step $1$: Factorise the polynomial to find its zeroes

Given polynomial, $3x^2-x-4$

First, split the middle term.

$3x^2-x-4=3x^2-4x+3x-4$

$=x(3x-4)+1(3x-4)$ [ taking the common terms out ]

$=\left(x+1\right)(3x-4)$

Zeroes can be found as

$\left(x+1\right)(3x-4)=0$

$$\begin{gathered} \Rightarrow x+1=0 \\ \Rightarrow x=-1 \end{gathered}$$

or,

$$\begin{gathered} \Rightarrow 3x-4=0 \\ \Rightarrow x=\frac{4}{3} \end{gathered}$$

Step $2$: Verify the relationship between the zeroes and the coefficients

Sum of the zeroes $=-1+\frac{4}{3}$

$$\begin{gathered} =\frac{1}{3} \\ =-\frac{b}{a} \end{gathered}$$

Product of zeroes $=-1\times \frac{4}{3}$

$$\begin{gathered} =-\frac{4}{3} \\ =\frac{c}{a} \end{gathered}$$

Hence, the zeroes of the polynomial are $-1$ and $\frac{4}{3}$.