Question

Choose the correct option from the given below options by solving the following question:
If $$\alpha ,\beta$$ are the zeros of the polynomial $$f\left(x\right)=x^2-p\left(x+1\right)-c=0$$ such that $$\left(\alpha +1\right)\left(\beta +1\right)=0$$, then $$c=?$$
A. $$1$$
B. $$0$$
C. $$-1$$
D. $$2$$

Answer 1

Hint: From the question we have zeros of the given polynomial and we have to find the value of $c$. By using the relation for sum of roots and product of roots with coefficients, we will get the required solution. We have to substitute the compared values in the relation to get the answer.

Complete step-by-step solution:
Given expression,
$$f\left(x\right)=x^2-p\left(x+1\right)-c=0$$
Given condition,
$$\left(\alpha +1\right)\left(\beta +1\right)=0$$
The variable we need to find,
$$c=?$$
Given $$\alpha$$ and $$\beta$$ are the zeros of the given polynomial.
Now, taking the given polynomial and expanding it, we get;
$$x^2-px-p-c=0$$
Now, bringing all the similar degree values to one term by taking common terms out, we get;
Comparing the above expression with the standard expression of a quadratic equation:
The standard equation for a quadratic equation$$=a{x}^2+bx+c=0$$
Now, comparing the equations, we get the coordinates of each like term;
$$a=1$$
$$b=-p$$
$$c=c-p$$
Now, taking the condition into consideration;
$$\left(\alpha +1\right)\left(\beta +1\right)=0$$
Expanding the equation, we get;
$$\alpha \beta +\alpha +\beta +1=0$$
Since, $$\alpha$$ and $$\beta$$ are the zeros of the given polynomial. Zeros of the polynomial also means that they are the roots of the polynomial.
According to the properties of the roots of polynomial,
Sum of roots $$=\alpha +\beta ={\displaystyle \frac{-b}{a}}$$ and
Product of the roots $$=\alpha \beta ={\displaystyle \frac{c}{a}}$$
Substituting the values, we got from the comparison in the above two expressions, we get;
Sum of roots $$={\displaystyle \frac{-b}{a}}={\displaystyle \frac{-\left(-p\right)}{1}}$$
Solving the expression and simplifying it, we get;
Sum of roots $$=p$$
Product of the roots $$={\displaystyle \frac{c}{a}}={\displaystyle \frac{c-p}{1}}$$
Solving the expression and simplifying it, we get;
Product of the roots $$=c-p$$
Prior, we have the equation,
$$\alpha \beta +\alpha +\beta +1=0$$
Now, substituting the known values, we get;
$$c-p+p+1=0$$
Solving the above equation, we get;
$$c+1=0$$
Taking the numerical values to one side and the variable side on one side, we get;
$$c=-1$$
Therefore, we have the value of $$c=-1$$.

Option C is the correct answer.

Note: We can observe from the solution of the question we get, a root of the quadratic equation is a real number $$x$$ which satisfies the equation. They are also the $$x$$- intercepts of the quadratic function that is the intersection between the graphs of the quadratic function with the $$x$$- axis.