Question

Find a quadratic polynomial with the given numbers ​as the sum and product of its zeroes respectively.​

$\frac{-1}{4},\frac{1}{4}$

Answer 1

Step $1$: Interpret the given data.
Let $\alpha$ and $\beta$ be the zeroes of the given polynomial. Given that ,
Sum of zeroes $=\alpha +\beta =-\frac{1}{4}$
Product of zeroes $=\alpha \beta =\frac{1}{4}$

Step $2$: Write the standard form of quadratic polynomial with zeroes.

We know that ,
If $\alpha$ and $\beta$ are the zeroes of any quadratic polynomial, the quadratic polynomial can be written directly as:
$x^2-(\alpha +\beta )x+\alpha \beta$

Step $3$: Put the given values in the standard form of quadratic polynomial.

Now, using the given values

$x^2-(\alpha +\beta )x+\alpha \beta$

$=x^2-(-\frac{1}{4})x+\left(\frac{1}{4}\right)$

$=\frac{4x^2+x+1}{4}$

$=4x^2+x+1$ [Multiplying or dividing the polynomial by a constant (except 0) does not change the zeroes]

Therefore, $4x^2+x+1$ is the quadratic polynomial.