In This Lesson What Are Polynomials? Polynomial Operations 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 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)
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 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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) 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: 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:
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)
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) 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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.