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Polynomials & Factoring

Break apart complex expressions into simple pieces — the key skill for solving higher-degree equations.

What Are Polynomials?

A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Polynomials are classified by their degree (highest power) and number of terms:

  • Monomial: 5x³ (one term)
  • Binomial: x² + 3 (two terms)
  • Trinomial: 2x² − 5x + 1 (three terms)
    • 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
Standard form: aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ⋯ + a₁x + a₀
Degree n, leading coefficient aₙ

The behavior of polynomials at large values — their end behavior — depends on the degree and leading coefficient. This becomes crucial when you study limits in calculus.

Polynomial Operations

Addition & Subtraction

Combine like terms (same variable and exponent):

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Example

(3x² + 2x − 5) + (x² − 4x + 7) = 4x² − 2x + 2

Multiplication

Distribute each term in the first polynomial across every term in the second (the "FOIL" method is a special case for two binomials):

Example: FOIL

(2x + 3)(x − 4) = 2x² − 8x + 3x − 12 = 2x² − 5x − 12

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Polynomial Long Division

Dividing polynomials works just like long division with numbers. This technique is essential for finding asymptotes of rational functions.

Example: (2x³ + 3x² − x + 5) ÷ (x + 2)

Result: 2x² − x + 1 with remainder 3

So: 2x³ + 3x² − x + 5 = (x + 2)(2x² − x + 1) + 3

Factoring Techniques

Factoring is the reverse of multiplication. It's the single most useful algebraic skill for solving equations.

1. Greatest Common Factor (GCF)

6x³ + 9x² = 3x²(2x + 3)

Always look for GCF first!

2. Difference of Squares

a² − b² = (a + b)(a − b)

Example:

25x² − 49 = (5x + 7)(5x − 7)

3. Perfect Square Trinomials

a² + 2ab + b² = (a + b)²
a² − 2ab + b² = (a − b)²

4. Trinomial Factoring (ac-method)

For ax² + bx + c, find two numbers that multiply to ac and add to b:

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Example: Factor 6x² + 11x + 3

ac = 18. Numbers that multiply to 18 and add to 11: 9 and 2

6x² + 9x + 2x + 3 = 3x(2x + 3) + 1(2x + 3) = (3x + 1)(2x + 3)

5. Sum/Difference of Cubes

a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)

6. Factor by Grouping

For polynomials with 4+ terms, group pairs and extract common factors:

Example: Factor x³ + x² + 2x + 2

Group: x²(x + 1) + 2(x + 1) = (x² + 2)(x + 1)

Factoring connects to many areas: the Fundamental Theorem of Arithmetic (unique prime factorization), partial fractions in integration, and characteristic polynomials in linear algebra.

Rational Expressions

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A rational expression is a fraction of two polynomials: P(x)/Q(x) where Q(x) ≠ 0. The techniques are identical to fraction arithmetic, but with polynomials. Factor first, then simplify.

Example: Simplify (x² − 9) / (x² + 5x + 6)

Factor: (x+3)(x−3) / (x+2)(x+3)

Cancel (x+3): (x−3)/(x+2), valid for x ≠ −3, x ≠ −2

Factor & Remainder Theorems

Remainder Theorem: If polynomial f(x) is divided by (x − c), the remainder is f(c).

Factor Theorem: (x − c) is a factor of f(x) if and only if f(c) = 0.

These theorems let you test potential roots by simple evaluation. Combined with the Rational Root Theorem — which says any rational root p/q must have p dividing the constant term and q dividing the leading coefficient — you can systematically find all rational roots of a polynomial.

The Factor Theorem is a bridge to the number-theoretic concept of divisibility, and the Rational Root Theorem connects to prime factorization.

Related Topics

Equations Use factoring to solve quadratic and higher-degree equations. Integrals Partial fraction decomposition — factoring's starring role in calculus. Prime Numbers Factoring integers mirrors factoring polynomials — same idea, different objects. Eigenvalues Factor the characteristic polynomial to find eigenvalues of a matrix.