Question

Find the factored form of the polynomial $ p(x) $ which has a simple zero at $ x = - 3 $ and a double zero at $ x = \dfrac{5}{4} $

Answer 1

Hint: The given question is related to the factorization of the polynomials: the zeroes of any polynomial expression, be it linear, quadratic or cubic, are the values at which the polynomial becomes equal to zero. For example the expression $ (x - 4)(x - 5) $ will become zero at the $ 4 $ and $ 5 $ . The concept of double zero is simple to the concept of zeroes; it is just that when there is a double zero it means that the factor is squared. The question can thus be solved by finding the factors of the polynomial by the use of these zeroes.

Complete step-by-step answer:
The given question asks us to find the factored form of the polynomial whose zeroes and double zeroes are given,
The factor corresponding to the zero value of $ x = - 3 $ gives us the factor as,
 $ x + 3 = 0 $ , so we have a factor of the polynomial $ P(x) $
The factor corresponding to the double zero at $ x = \dfrac{5}{4} $ , will be a squared factor , it will be given by,
 $ \Rightarrow x = \dfrac{5}{4} $
 $ \Rightarrow 4x - 5 = 0 $
But since it is double zero it will be squared. So the factor is
 $ {(4x - 5)^2} $
The factored form of the polynomial using these factors will be,
 $ P(x) = (x + 3){(4x - 5)^2} $
So, the correct answer is “ $ P(x) = (x + 3){(4x - 5)^2} $ ”.

Note: In case the polynomial itself was asked we would have further solved the factored form given above to get a given polynomial in the form of a cubic expression. A polynomial expression which are linear, quadratic or cubic
Have degrees as $ 1,2 $ and $ 3 $ respectively and the number of factors is also the same as their degree, so a linear equation will have only $ 1 $ factor while a cubic equation has $ 3 $ factors, they may be real or imaginary.