Polynomials
Grade Class 10
Question:
<p>If α, β are the zeros of the polynomial <em>p</em>(<em>x</em>) = 4<em>x</em><sup>2</sup> + 3<em>x</em> + 7, then <span class="math-tex">\(\frac{1}{\alpha}+\frac{1}{\beta}\)</span> is equal to</p>
<p style="display:inline"><span class="math-tex">\(-\frac{7}{3}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{7}{3}\)</span></p>
<p style="display:inline"><span class="math-tex">\(-\frac{3}{7}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{3}{7}\)</span></p>
Step-by-Step Solution
Key Concept: Evaluate symmetric expressions of roots by rewriting them in terms of the sum (-b/a) and product (c/a) of the zeros.
<p>Since <span class="math-tex">\(\alpha\)</span> and <span class="math-tex">\(\beta\)</span> are the zeros of the quadratic polynomial <span class="math-tex">\(p(x)=4 x^{2}+3 x+7\)</span></p>
<p><span class="math-tex">\(\alpha+\beta=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}\)</span> <span class="math-tex">\(=\frac{-3}{4}\)</span></p>
<p><span class="math-tex">\(\alpha \beta=\frac{\text { Constant term }}{\text { coefficient of } x^{2}}\)</span> <span class="math-tex">\(=\frac{7}{4}\)</span></p>
<p>Now, <span class="math-tex">\(\frac{1}{\alpha}+\frac{1}{\beta}\)</span> <span class="math-tex">\(=\frac{\beta+\alpha}{\alpha \beta}\)</span> <span class="math-tex">\(=\frac{\frac{-3}{4}}{\frac{7}{4}}\)</span> <span class="math-tex">\(=\frac{-3}{4} \times \frac{4}{7}\)</span> <span class="math-tex">\(=\frac{-3}{7}\)</span><br />
Thus, the value of <span class="math-tex">\(\frac{1}{a}+\frac{1}{\beta}\)</span> is <span class="math-tex">\(\frac{-3}{7}\)</span>.</p>
Correct Answer: C