Sets, Relations & Functions
General
Grade 11
Question:
<p>Let <span class="math-inline">\(f\)</span> be a degree-3 polynomial with <span class="math-inline">\(f(k)=-2/k\)</span> for <span class="math-inline">\(k=2,3,4,5\)</span>. Find <span class="math-inline">\(52-10f(10)\)</span>.</p>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>Define P(x)=xf(x)+2. P(2)=P(3)=P(4)=P(5)=0. P(x)=λ(x-2)(x-3)(x-4)(x-5). P(0)=2=λ·120 → λ=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 λ must be found from P(0).</div><div class="key-concept"><strong>Key Concept:</strong> f(k)=-2/k → multiply by x → quartic with known roots</div></div>
Correct Answer: 26