In This Lesson What Are Polynomials? Polynomial Operations Factoring Techniques Rational Expressions Factor & Remainder Theorems 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:
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 Monomial: 5x³ (one term)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 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 (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) 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) 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 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)
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 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 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 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
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)
6. Factor by Grouping 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 For polynomials with 4+ terms, group pairs and extract common factors:
Example: Factor x³ + x² + 2x + 2 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 Group: x²(x + 1) + 2(x + 1) = (x² + 2)(x + 1)
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) Factor: (x+3)(x−3) / (x+2)(x+3)
Cancel (x+3): (x−3)/(x+2) , valid for x ≠ −3, x ≠ −2
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 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.