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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)
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):

Example

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

(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

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)

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

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:

Example: Factor 6x² + 11x + 3

zTOBq5Puuk+KzRCmZydqMAq5UY862rKjZvnbNS0m+wYAo9huqnMptmDy4udcF5bAfonW2+VccYYzJ+sztV7KLvALkPs3iovRhqnXAYz7bk8FSixtTLH9gbWhS5upERBw2nGxNsh5GnyDoFFjUDuZSFbY1r64LJ4ZRb/otr4kMJEsQXg6U1zAlE/AGwfRj0Lth1OF7bqG+tCEGTCKIEe0EI4T9HlpOPNwk25KV1B0TouEBvIgxInMNt0g3B5s3uBc7BYJwVgurAO9b8BKfSwUEmfagtKe4/5aSUa+yjw0LP1Lguj/dt6d6+KTjI+tKY2IkmeSTIvbksW/VDH5PNNNg4RNlcMBj3p1lyK9IwTPTo4e8UfoSRdIKUBJSsTMYbj9nKyPDu7l3iA3Uni4IhwT4yfftif3boczJ7FkiRv3QXnghjklHOBav0mdyCrBQ9OSfvPUfFsMrV9TIlUy7A3GdaF3Iz/a1akMkoieFCPGq4DT7PekMQjTN516kPAaBqt8EGRR0XZlQea3mh+FZCesmiMpUvUvG097ly+KkAYbksDbHFxlZ394zP++edtbRC174gXQISOBRyhM/8UzJK4BM8bkc3jcENrw7VmLgyqks78wyFVvyGRCpdikPnn+RZZv3CXxXeWBl4zy4Wi2g9OmTb1M2h1cBzZV7WQ/kNn4uYIGGNLL+Rw72XiAkl+rgL3+YVN1Oq6vYYCOnO22ChCeXKIT0eybBFU36AYkO3kRLTxgoc1B/IgJCsHjv6lglaAwPHO8L1JIcVJZWiLNhIEOoGC5VhNMP4iZp2ANPH/lEy44iXErLEOEvFhmlwWxyMEGfYsQCVMGi91aVT/qdY0S

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:

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

Example: Factor x³ + x² + 2x + 2

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

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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

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)

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

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.
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