AMC 10 · 2021 · #15

Grade 8 algebra

Archetype Arithmetic Expression Decomposition └ Sub-archetype Multi-Segment Path or Count Sum

polynomial-factoringpolynomial-substitutionrecursive-sequenceexponents identify-subproblemseasier-related-problem ↑ Prerequisites: polynomial-factoringexponents

📏 Medium solution 💡 3 insights

Problem

A real number x satisfies x +

\frac{1}{x}

= √(5). Find the value of x¹¹ - 7 x⁷ + x³.

Pick an answer.

(A)

~-1

(B)

(C)

(D)

(E)

$~\sqrt{5}$

AMC 10 2021 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

Tool #7 (Subproblems): the target x¹¹ - 7x⁷ + x³ factors as x³(x⁸ - 7x⁴ + 1), splitting the work into 'find x⁴ in simple form' and 'evaluate the bracket'. Tool #9 (Easier Problem): instead of working with the messy √(5), square x + $\frac{1}{x}$ = √(5) to get a polynomial relation, then multiply by x² to clear denominators. The result x⁴ = 3x² - 1 is the easier version. Tool #13 (Algebra): substitute u = x⁴ = 3x² - 1 into u² - 7u + 1 and watch all the x² terms cancel to 0. Tool #3 (Eliminate) matches 0 to choice (B).

1STEP 1

Factor to expose structure: x¹¹ - 7x⁷ + x³ = x³(x⁸ - 7x⁴ + 1) = x³(u² - 7u + 1) with u = x⁴.

x¹¹ - 7x⁷ + x³ = x³ (x⁸ - 7 x⁴ + 1) = x³ (u² - 7 u + 1) where u = x⁴

2STEP 2

Square the given to kill √(5): (x + $\frac{1}{x}$ )² = 5, so x² + 2 + $\frac{1}{x²}$ = 5, giving x² + $\frac{1}{x²}$ = 3.

x² +

\frac{1}{x²}

= 3

3STEP 3

Multiply by x² to clear denominators: x⁴ + 1 = 3x², i.e. x⁴ = 3x² - 1 — u = x⁴ in terms of x².

u = x⁴ = 3 x² - 1

4STEP 4

Expand u² = (3x² - 1)² and reduce the leftover x⁴ again to get u² = 21x² - 8.

u² = 9 x⁴ - 6 x² + 1 = (27 x² - 9) - 6 x² + 1 = 21 x² - 8

5STEP 5

Substitute u² = 21x² - 8 and u = 3x² - 1: (21x² - 8) - 7(3x² - 1) + 1, the x² terms cancel, so the bracket = 0.

u² - 7 u + 1 = (21 x² - 8) - (21 x² - 7) + 1 = 0

6STEP 6

Combine: x¹¹ - 7x⁷ + x³ = x³·(u² - 7u + 1) = x³·0 = 0 — no value of x needed, so 0 → (B).

x¹¹ - 7 x⁷ + x³ = x³ · 0 = 0 → (B)

Answer

Independent check: from x +

\frac{1}{x}

= √(5), the quadratic x² - √(5) x + 1 = 0 gives x =

\frac{√(5) ± 1}{2}

(a positive root of value

\frac{√(5) + 1}{2}

≈ 1.618, the golden ratio φ). Using φ: φ² = φ + 1 ≈ 2.618, φ⁴ = (φ+1)² = φ² + 2φ + 1 ≈ 6.854. And 3φ² - 1 = 3(2.618) - 1 = 6.854. So x⁴ = 3x² - 1 holds. The bracket x⁸ - 7x⁴ + 1 = (3x² - 1)² - 7(3x² - 1) + 1 = 0 confirms the value 0 on the nose. Answer (B) makes sense.

💡Key takeaway

This AMC 10 problem only needs Grade 8 exponent rules you already know! Factor the target: x¹¹ - 7x⁷ + x³ = x³(x⁸ - 7x⁴ + 1). Squaring x + $\frac{1}{x}$ = √(5) gives x² + $\frac{1}{x²}$ = 3, so x⁴ = 3x² - 1. Plug into the bracket — every x² cancels and the bracket equals 0. So the whole expression is 0, answer (B).

More like this