Question

Find the zeroes of the following quadratic polynomials and verify the relationship between zeroes and their coefficients.

$3x^2-x-4$

Answer 1

Definition of Polynomial:

An expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).

Finding the zeroes:

$$\begin{gathered} p\left(x\right)=3x^2-x-4 \\ p\left(x\right)=3x^2-4x+3x-4 \\ p\left(x\right)=x(3x-4)+1(3x-4) \\ p\left(x\right)=(x+1)(3x-4) \end{gathered}$$

Therefore, $$\begin{gathered} x+1=0 \\ x=-1 \end{gathered}$$ or $$\begin{gathered} 3x-4=0 \\ 3x=4 \\ x=\frac{4}{3} \end{gathered}$$

The two zeroes of the polynomial are $-1and\frac{4}{3}$

Verifying the relationship between the zeroes and their coefficients.

Let the zeroes be $\alpha and\beta$

Sum of zeroes $=\alpha +\beta =\frac{-b}{a}=\frac{-coefficientofx}{coefficientof{x}^2}=\frac{-(-1)}{3}=\frac{1}{3}$

Sum of zeroes$$\begin{gathered} =-1+\frac{4}{3} \\ =\frac{-3+4}{3} \\ =\frac{1}{3} \end{gathered}$$

Product of zeroes$=\alpha \beta =\frac{c}{a}=\frac{cons\tan tterm}{coefficientof{x}^2}=\frac{-4}{3}$

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

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