Polynomials
Grade Class 10
Question:
<p>The number of polynomials having zeroes -1 and 2 is:</p>
<p style="display:inline">infinite</p>
<p style="display:inline">exactly 2</p>
<p style="display:inline">at most 2</p>
<p style="display:inline">only 1</p>
Step-by-Step Solution
Key Concept: A polynomial with given roots is defined as k(x² - (sum)x + product), where the arbitrary non-zero constant k allows for infinitely many such polynomials.
<p>Let <span class="math-tex">$\alpha$</span> = -1 and <span class="math-tex">$\beta$</span> = 2<br />
<span class="math-tex">$\alpha + \beta$</span> = -1 + 2 = 1<br />
<span class="math-tex">$\alpha \beta$</span> = (-1) (2) = -2<br />
Required polynomial = k(x<sup>2</sup> - <span class="math-tex">$(\alpha + \beta)$</span>x + <span class="math-tex">$\alpha \beta$</span>)<br />
= k(x<sup>2</sup> - x - 2)<br />
There are infinite values of k is possible.<br />
Hence, number of Polynomial whose zeroes are -1 and 2 are infinite.</p>
Correct Answer: A