🎯 Polynomials & Rational Functions

Polynomials and rational functions are central to AMC problems. Master factoring, Vieta’s formulas, and asymptote analysis.

🎯 Key Ideas

Polynomials: Sums of power terms $p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$

Rational Functions: Ratios of polynomials $r(x) = \frac{p(x)}{q(x)}$ where $q(x) \neq 0$

Vieta’s Formulas: Relate polynomial coefficients to sums and products of roots.

🔢 Polynomial Basics

Degree and Leading Coefficient

• Degree: Highest power of $x$ with non-zero coefficient
• Leading Coefficient: Coefficient of highest degree term
• Constant Term: Coefficient of $x^0$ term

Example: $p(x) = 3x^4 - 2x^2 + 5x - 1$

• Degree: 4
• Leading coefficient: 3
• Constant term: -1

✂️ Factoring Techniques

Common Patterns

• Difference of squares: $a^2 - b^2 = (a-b)(a+b)$
• Perfect squares: $a^2 \pm 2ab + b^2 = (a \pm b)^2$
• Sum/difference of cubes: $a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$
• Grouping: Factor common terms from groups

Example: Factor $x^3 - 8$

• Recognize as difference of cubes: $x^3 - 2^3$
• Apply formula: $(x-2)(x^2 + 2x + 4)$

Pitfall: Incomplete Factoring

Always check if further factoring is possible. Look for common factors first.

🎯 Vieta’s Formulas

For polynomial $ax^2 + bx + c = 0$ with roots $r_1$ and $r_2$:

• Sum of roots: $r_1 + r_2 = -\frac{b}{a}$
• Product of roots: $r_1 r_2 = \frac{c}{a}$

For cubic $ax^3 + bx^2 + cx + d = 0$ with roots $r_1, r_2, r_3$:

• Sum: $r_1 + r_2 + r_3 = -\frac{b}{a}$
• Sum of products: $r_1 r_2 + r_1 r_3 + r_2 r_3 = \frac{c}{a}$
• Product: $r_1 r_2 r_3 = -\frac{d}{a}$

Example: If $x^2 - 5x + 6 = 0$ has roots $r$ and $s$, find $r^2 + s^2$

Solution:

1. From Vieta’s: $r + s = 5$ and $rs = 6$
2. Use identity: $r^2 + s^2 = (r+s)^2 - 2rs$
3. Substitute: $r^2 + s^2 = 5^2 - 2(6) = 25 - 12 = 13$

🔄 Remainder and Factor Theorems

Remainder Theorem: When polynomial $p(x)$ is divided by $(x-a)$, the remainder is $p(a)$.

Factor Theorem: $(x-a)$ is a factor of $p(x)$ if and only if $p(a) = 0$.

Example: Find remainder when $x^3 - 2x^2 + 3x - 1$ is divided by $(x-2)$

Solution:

• By Remainder Theorem: remainder = $p(2) = 2^3 - 2(2^2) + 3(2) - 1 = 8 - 8 + 6 - 1 = 5$

📊 Rational Functions

Asymptotes

• Vertical asymptotes: Where denominator is zero (but numerator isn’t)
• Horizontal asymptotes: Compare degrees of numerator and denominator
• Oblique asymptotes: When degree of numerator = degree of denominator + 1

Asymptote Rules

Example: Find asymptotes of $f(x) = \frac{x^2 - 1}{x^2 - 4}$

Solution:

1. Vertical asymptotes: Set denominator = 0
   • $x^2 - 4 = 0$ → $x = \pm 2$
2. Horizontal asymptote: Degrees are equal (both 2)
   • $y = \frac{1}{1} = 1$

🎯 AMC-Style Worked Example

Problem: If $x^3 - 6x^2 + 11x - 6 = 0$ has roots $a, b, c$, find $a^2 + b^2 + c^2$.

Solution:

1. Apply Vieta’s formulas:

   • $a + b + c = 6$ (sum of roots)
   • $ab + ac + bc = 11$ (sum of products)
   • $abc = 6$ (product)

2. Use the identity: $a^2 + b^2 + c^2 = (a+b+c)^2 - 2(ab+ac+bc)$

3. Substitute: $a^2 + b^2 + c^2 = 6^2 - 2(11) = 36 - 22 = 14$

Answer: 14

🔍 Common Traps & Fixes

Trap: Wrong Vieta’s formula signs

Fix: Remember the negative sign: sum of roots = $-\frac{b}{a}$

Trap: Confusing holes and asymptotes

Fix: Holes occur when both numerator and denominator have the same zero; asymptotes when only denominator is zero.

Trap: Forgetting to check domain restrictions

Fix: Always identify where rational functions are undefined.

Trap: Incomplete factoring

Fix: Check for common factors and use multiple techniques.

📋 Quick Reference

Essential Factoring Formulas

• $a^2 - b^2 = (a-b)(a+b)$
• $a^2 \pm 2ab + b^2 = (a \pm b)^2$
• $a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$
• $a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3$

Vieta’s for Quadratic

• Sum: $r_1 + r_2 = -\frac{b}{a}$
• Product: $r_1 r_2 = \frac{c}{a}$

Common Identities

• $r_1^2 + r_2^2 = (r_1 + r_2)^2 - 2r_1 r_2$
• $r_1^3 + r_2^3 = (r_1 + r_2)^3 - 3r_1 r_2(r_1 + r_2)$

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