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

$\left(\frac{21}{8}\right),\left(\frac{5}{16}\right)$

Answer 1

Step 1: Solve for equation of the quadratic polynomial

Given,

1. Sum of zeros $=\frac{21}{8}$.
2. Product of zeros$=\frac{5}{16}$.

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

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

$$\begin{gathered} \therefore x^2-\left(\frac{21}{8}\right)x+\left(\frac{5}{16}\right)=0 \\ \Rightarrow 16x^2-42x+5=0 \end{gathered}$$

Step 2: Factorize the polynomial

$$\begin{gathered} 16x^2-42x+5=0 \\ \Rightarrow 16x^2-2x-40x+5=0 \\ \Rightarrow 2x\left(8x-1\right)-5\left(8x-1\right)=0 \\ \Rightarrow \left(2x-5\right)\left(8x-1\right)=0 \\ \Rightarrow x=\frac{5}{2},\frac{1}{8} \end{gathered}$$

Hence, the zeroes of the given polynomial are $\frac{5}{2},\frac{1}{8}$