Question

Find a quadratic polynomial whose zeroes are $ - 4 $ and $ - 5 $ ?

Answer 1

Hint: The zeroes of a polynomial are the points at which a given polynomial becomes zero. The zeroes of a polynomial help in finding the factors of that polynomial equation and can help to find the polynomial itself. The zeroes of a polynomial of the type say
\[(x - a)(x - b) = 0\]
are present at
 $ x = a_{}^{} $ and at $ x = n $
We call them zero because when we put $ x = a\,{\text{ or }}x = b $ in the equation we get a zero.
The same process can be used in reverse to find the polynomial when the zeroes are given.

Complete step-by-step answer:
We have to find the polynomial whose zeroes are given at the points $ - 4, - 5 $ .
For the zero at $ - 4 $ , we can say the factor will be
 $ x = - 4 $
Which yields us with the factor of the polynomial as,
 $ x + 4 = 0 $ . This will be the first factor of the polynomial.
Similarly we will find the other factor of the polynomial. The factor for the polynomial at zero of $ - 5 $ is
 $ x = - 5 $ ,
Which gives us the factor as,
 $ x + 5 = 0 $
This is our second factor.
Now since the equation is a quadratic equation, it will have only two values of $ x $ and therefore the polynomial
 $ P(x) $ can be written as,
 $ \Rightarrow P(x) = (x + 5)(x + 4) $
 $ \Rightarrow P(x) = {x^2} + 5x + 4x + 20 $
 $ \Rightarrow P(x) = {x^2} + 9x + 20 $
Which is the required polynomial.
So, the correct answer is “ $ P(x) = {x^2} + 9x + 20 $ ”.

Note: In case the question said there is a double zero present the double zero in a polynomial would have meant that the factor is a square, so say a polynomial has a double zero at $ 4 $ , so we can write that
 $ {(x - 4)^2} $ will be the factor of that polynomial.