Sets, Relations & Functions
General
Grade 11
Question:
<p>Let <span class="math-inline">\(P(x) = x^6 + ax^5 + bx^4 + cx^3 + dx^2 + ex + f\)</span> be a polynomial such that <span class="math-inline">\(P(1)=1, P(2)=2, P(3)=3, P(4)=4, P(5)=5, P(6)=6\)</span>. Find <span class="math-inline">\(P(7)\)</span>.</p>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p><strong>Key Idea:</strong> Subtract the identity. Let <span class="math-inline">$Q(x)=P(x)-x$</span>, which has roots <span class="math-inline">$1,2,3,4,5,6$</span>.</p><p><strong>Step 1:</strong> <span class="math-block">$$Q(x) = (x-1)(x-2)(x-3)(x-4)(x-5)(x-6)$$</span></p><p><strong>Step 2:</strong> <span class="math-block">$$P(7) = 7 + 6\cdot5\cdot4\cdot3\cdot2\cdot1 = 7 + 720 = 727$$</span></p><p><strong>Answer: 727</strong></p><div class="trap-box"><strong>Trap:</strong> Never try to solve for all 6 coefficients. That is pure computation with no insight.</div><div class="key-concept"><strong>Key Concept:</strong> Polynomial construction by subtracting identity — convert interpolation into roots problem</div></div>
Correct Answer: 727