Question

Form a quadratic polynomial whose zeroes are $ 7 + \sqrt 5 $ and $ 7 - \sqrt 5 . $

Answer 1

Hint: A quadratic polynomial is polynomial of degree $ 2 $ . Every quadratic polynomial can be written in terms of sum and product of its roots.

Complete step-by-step answer:
Given, zeroes are $ 7 + \sqrt 5 $ and $ 7 - \sqrt 5 $
To form a quadratic polynomial, at first, we have to find ‘sum of zeroes’
sum of zeroes $ = 7 + \sqrt 5 + 7 - \sqrt 5 $
Cancelling the negative and positive $ \sqrt 5 $ we get
Sum of zeroes $ = 7 + 7 $
 $ \Rightarrow $ Sum of zeroes $ = 14 $ . . . (1)
Now, we have to find ‘product of zeroes’
Product of zeroes $ = \left( {7 + \sqrt 5 } \right)\left( {7 - \sqrt 5 } \right) $
We know that,
 $ (a + b)(a - b) = {a^2} - {b^2} $
 $ \Rightarrow $ Product of zeroes $ = {\left( 7 \right)^2} - {\left( {\sqrt 5 } \right)^2} $
 $ \Rightarrow $ Product of zeroes $ = 49 - 5 $
 $ \Rightarrow $ Product of zeroes $ = 44 $ . . . (2)
Now, a quadratic equation in terms of sum and product of roots can be written as
 $ {x^2} - Sx + P = 0 $ . . . (3)
Where, S is sum of roots
And P is product or roots.
Therefore, by using equations (1), (2) and (3)
We can write the quadratic polynomial as
 $ {x^2} - 14x + 44 $
Therefore, the required quadratic polynomial is $ {x^2} - 14x + 44 $

Note: The standard form is quadratic polynomial is $ a{x^2} + bx + c = 0. $ where $ a \ne 0 $
In this polynomial, sum of roots $ = - \dfrac{b}{a} $
And product of roots $ = \dfrac{c}{a} $
Zeros and roots are the same thing. Some books use the term zeros and some use roots. Do not get confused with it. Both the words represent the same thing.