Sets, Relations & Functions
General
Grade 11
Question:
<p>Let \(f\) be a degree-3 polynomial with \(f(k)=-2/k\) for \(k=2,3,4,5\). Find \(52-10f(10)\).</p>
Step-by-Step Solution
<div class="solution"><p>Define P(x)=xf(x)+2. P(2)=P(3)=P(4)=P(5)=0. P(x)=\lambda(x-2)(x-3)(x-4)(x-5). P(0)=2=\lambda·120 \to \lambda=1/60. P(10)=10f(10)+2=(8·7·6·5)/60=28. So 10f(10)=26, answer=52-26=26.</p><div class="trap-box"><strong>Trap:</strong> The scaling \lambda must be found from P(0).<div class="key-concept"><strong>Key Concept:</strong> f(k)=-2/k \to multiply by x \to quartic with known roots
Correct Answer: 26