Question

Form a quadratic polynomial f(x) with sum and product of zeroes are 2 and - 3/5 respectively.​

Answer 1

The Answer is

The polynomial f(x) = 5x² - 10x - 3, whose sum of zeroes is 2, and product of the zeroes is (-3/5).

Full explanation:

The sum of zeroes is 2 (given)

The product of zeroes is (-3/5) (given)

TO FIND:

The quadratic polynomial f(x), where sum of the zeroes and the product of the zeroes will be 2, (-3/5) respectively.

The formula for a quadratic polynomial is

⇒ K[x² - (sum of the zeroes)x + (product of zeroes)]

So,

Putting the given values in the above formula, we will get the required polynomial.

→ K [ x² - (2)x + (-3/5) ]

→ K[ x² - 2x - (3/5) ]

→ K[ (5x² - 10x - 3)/5 ]

∵ By taking LCM = 5

→ 5 [ (5x² - 10x - 3)/5 ]

∵ Taking K (constant) = 5 so, 5 will be cancelled and we will get,

→ 5x² - 10x - 3

Thus,

The f(x) = 5x² - 10x - 3

Answer 2

Given:

The sum of zeroes is 2

The product of zeroes is (-3/5)

 

To find:

A quadratic polynomial whose sum is the zeroes will be 2 and the product of the zeroes will be (-3/5)

So,

We know to find a quadratic polynomial

$\boxed{\red{\sf{\implies k(x^{2}-(sum\ of\ zeroes)x + Product\ of\ zeroes)}}}$

Here k is a constant.

Now,

$\implies k(x^{2}-(2)x +( \frac{-3}{5}))$

$\implies k(x^{2}-2x - \frac{3}{5})$

$\implies k( \frac{5x^{2}-10x -3}{5})$

$\implies \boxed{\pink{\tt{5x^{2}-10x -3}}}}$

Hence,

The required polynomial f(x) = 5x² - 10 - 3

- - -

Verify

The sum of zeroes $\tt =\frac{coefficient\ of\ x}{coefficient\ of\ x^{2}}=\frac{-b}{a} = \frac{-(-10)}{5} = \frac{10}{5} = 2$

And

The product of zeroes $\tt =\frac{constant\ term}{coefficient\ of\ x^{2}}=\frac{c}{a} = \frac{-3}{5}$

Thus, verified also.