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:
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
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 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 (2x + 3)(x − 4) = 2x² − 8x + 3x − 12 = 2x² − 5x − 12
Polynomial Long Division 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 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
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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: 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 25x² − 49 = (5x + 7)(5x − 7)
3. Perfect Square Trinomials
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)
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)
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
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.