Question

For each of the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorization

$\left(\frac{-8}{3}\right),\left(\frac{4}{3}\right)$

Answer 1

Step 1: Solve for equation of the quadratic polynomial

Given,

1. Sum of zeros $=\frac{-8}{3}$
2. Product of zeros$=\frac{4}{3}$

We know that, the equation of the quadratic polynomial can be written as,

$x^2-$(sum of zeros)$+$(product of zeros)$=0$

$$\begin{gathered} \therefore x^2-\left(\frac{-8}{3}\right)x+\left(\frac{4}{3}\right)=0 \\ \Rightarrow 3x^2+8x+4=0 \end{gathered}$$

Step 2: Factorize the polynomial

$$\begin{gathered} 3x^2+8x+4=0 \\ \Rightarrow x\left(3x+2\right)+2\left(3x+2\right)=0 \\ \Rightarrow \left(x+2\right)\left(3x+2\right)=0 \\ \Rightarrow x=-2,\frac{-2}{3} \end{gathered}$$

Hence, the zeroes of the given polynomial are $-2,\frac{-2}{3}$