Question:
<p>If p, q are the zeroes of the polynomial f(x) = x<sup>2 </sup>+ k(x - 1) - c, then (p - 1)(q - 1) is equal to ________.</p>
<p style="display:inline">1 + c</p>
<p style="display:inline">c - 1</p>
<p style="display:inline">c</p>
<p style="display:inline">1 - c</p>
Step-by-Step Solution
Key Concept: The problem is solved by rewriting the polynomial in standard quadratic form and applying Vieta's formulas to relate the roots to the coefficients.
<p>Given equation is x<sup>2</sup> + k(x - 1) - c<br />
= x<sup>2</sup> + kx - (k + c)<br />
Since, p and qare the zeroes,<br />
<span class="math-tex">$\therefore$</span>p + q = -k and pq = -(k + c)<br />
Now, (p - 1) (q- 1) = pq- q- p + 1<br />
= pq- (p + q) + 1 = -(k + c) - (-k) + 1<br />
= -k - c + k + 1 = 1 - c</p>
Correct Answer: D