Quadratic Equations
Polynomials with restricted coefficients
Grade None
Question:
<p>For polynomials of the form \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\) with \(a_i \in \{-1, 1\}\), \((i = 0, 1, 2, \ldots, n)\) which has all roots real, then</p>
<p>The maximum value of \(n\) is less than 7</p>
<p>The maximum value of \(n\) is more than 5</p>
<p>Number of such polynomials is less than 10</p>
<p>Number of such polynomials is more than 5</p>
Step-by-Step Solution
Key Concept: For a polynomial with coefficients in {-1, 1} to have all real roots, severe constraints exist on its structure. The discriminant and Vieta's formulas combined with the restricted coefficient set severely limit which polynomials can achieve this property.
<p><strong>Step 1: Analyze quadratic case</strong><br/>For quadratic ax² + bx + c with a,b,c ∈ {-1,1}, discriminant Δ = b² - 4ac must be ≥ 0.</p><p><strong>Step 2: Check all cases</strong><br/>• If a = 1, c = 1: Δ = b² - 4. For real roots: b² ≥ 4, so b ∈ {-1,1} gives Δ ∈ {-3, -2}. Not real roots.<br/>• If a = 1, c = -1: Δ = b² + 4 ∈ {5, 8}. Neither is perfect square (except specific checks).<br/>• If a = -1, c = 1: Δ = b² - 4(-1) = b² + 4. Same as above.<br/>• If a = -1, c = -1: Δ = b² + 4 ∈ {5, 8}. Not perfect squares.</p><p><strong>Step 3: Higher degree polynomials</strong><br/>For cubic and higher degrees with all ±1 coefficients, polynomials like (x-1)³ = x³ - 3x² + 3x - 1 don't have coefficients in {-1,1}. Monic polynomials with integer roots and coefficient restrictions are extremely limited.</p><p><strong>Step 4: Determine valid statements</strong><br/>Options A, C, D likely relate to the fact that very few such polynomials exist with all real roots. Linear polynomials (ax + b) always have real roots. Higher degrees with full ±1 coefficients having all real roots form a severely restricted set (possibly empty or nearly empty for degrees ≥ 2).</p><p>∴ Answer: ACD</p>
Correct Answer: ACD