Question:
<p>If α, β are the zeros of polynomial f(x) = x<sup>2</sup> − p (x + 1) − c, then (α + 1) (β + 1) =</p>
<p style="display:inline">c − 1</p>
<p style="display:inline">c</p>
<p style="display:inline">1 − c</p>
<p style="display:inline">1 + c</p>
Step-by-Step Solution
Key Concept: Express the polynomial in standard form to accurately determine the sum and product of zeros using Vieta's formulas.
<p>Since <span class="math-tex">\(\alpha\)</span> and <span class="math-tex">\(\beta\)</span> are the zeros of quadratic polynomial <span class="math-tex">\(f(x)=x^{2}-p(x+1)-c\)</span><br />
<span class="math-tex">\(=x^{2}-p x-p-c\)</span><br />
<span class="math-tex">\(\alpha+\beta=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}\)</span><br />
<span class="math-tex">\(=-\left(\frac{-p}{1}\right)\)</span> = p<br />
<span class="math-tex">\(\alpha \times \beta=\frac{\text { Constant term }}{\text { Coefficient of } x^{2}}\)</span><br />
<span class="math-tex">\(=\frac{-p-c}{1}\)</span> = <span class="math-tex">\(-p-c\)</span><br />
We have<br />
<span class="math-tex">\((\alpha+1)(\beta+1)\)</span><br />
<span class="math-tex">\(=\alpha \beta+\beta+\alpha+1\)</span><br />
<span class="math-tex">\(=\alpha \beta+(\alpha+\beta)+1\)</span><br />
<span class="math-tex">\(=-p-c+(p)+1\)</span><br />
<span class="math-tex">\(=-c+1\)</span><br />
= 1 - c<br />
The value of <span class="math-tex">\((\alpha+1)(\beta+1)\)</span> is 1 - c.</p>
Correct Answer: C