In This Lesson What Are Polynomials? Polynomial Operations Factoring Techniques Rational Expressions 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 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
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 .
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 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
Factoring Techniques 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 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 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)
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)
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 Cancel (x+3): (x−3)/(x+2) , valid for x ≠ −3, x ≠ −2
Factor & Remainder Theorems 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 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.