If α and β are the zeroes of the polynomial \(x^2+5x+c\), and \( α-β=3\), then \(c=\)?          a)  0                b) 1                 c) 4                       d) 5

Answer: Option c. 4 Solution: Step 1: Use the concepts of relationship between roots and coefficients of quadratic polynomial. Given polynomial is \(x^2+5x+c\), by comparing it standard quadratic polynomial \(ax^2+bx+c\), we get \(a=1, \quad b=5, \quad c=c\) \( \alpha+\beta= \frac{-b}{a}=\frac{-5}{1} = -5 \)            ....(1) \(\alpha \beta= \frac{c}{a}=\frac{c}{1} =c\)              ....(2) \(\alpha-\beta= 3\)     .....(3)    [given] Step 2: Find the value of α and β By adding (1) and (2), we get \(\alpha+\beta +\alpha-\beta= -5+3\) \(\Rightarrow 2\alpha = -2\) \(\Rightarrow \alpha = \frac{-2}{2}=-1\) Put the value of \(α= -1\)  in (1), we get \(-1+\beta= -5\) \(\Rightarrow \beta= -5+1\) \(\Rightarrow \beta= -4\) Step 3: Use the concept of product of roots of quadratic polynomial \(∵  \quad \alpha\times \beta = \frac{c}{a}\) \(\therefore  \quad \alpha\times \beta = \frac{c}{1}=c\) Substitute the value of \(\alpha\) and \(\beta\), we get \(c=-1 \times -4\) \(\Rightarrow c=4\) Hence, the value of c is 4.