Applications of Derivatives
Real Roots of Polynomials
Grade 12
Question:
<p><strong>Example 53:</strong> If <span>f(x)</span> is a polynomial of degree 5 with real coefficients such that <span>f(|x|) = 0</span> has 8 real roots, then <span>f(x) = 0</span> has</p>
<p>(a) 4 real roots</p>
<p>(b) 5 real roots</p>
<p>(c) 3 real roots</p>
<p>(d) nothing can be said</p>
Step-by-Step Solution
Key Concept: A polynomial of odd degree must have an odd number of real roots. The even number of roots from f(|x|) implies one additional real root must exist.
<p><strong>Step 1:</strong> Given that <span>f(|x|) = 0</span> has 8 real roots.</p><p><strong>Step 2:</strong> This means <span>f(x) = 0</span> has 4 positive roots.</p><p><strong>Step 3:</strong> Since <span>f(x)</span> is a polynomial of degree 5, <span>f(x)</span> cannot have an even number of real roots.</p><p><strong>Step 4:</strong> Therefore, <span>f(x)</span> has all five roots real: four positive and one negative.</p><p>∴ Answer is (b) 5 real roots.</p>
Correct Answer: B