Break apart complex expressions into simple pieces — the key skill for solving higher-degree equations.
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:
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.
Combine like terms (same variable and exponent):
(3x² + 2x − 5) + (x² − 4x + 7) = 4x² − 2x + 2
Distribute each term in the first polynomial across every term in the second (the "FOIL" method is a special case for two binomials):
(2x + 3)(x − 4) = 2x² − 8x + 3x − 12 = 2x² − 5x − 12
Dividing polynomials works just like long division with numbers. This technique is essential for finding asymptotes of rational functions.
Result: 2x² − x + 1 with remainder 3
So: 2x³ + 3x² − x + 5 = (x + 2)(2x² − x + 1) + 3
Factoring is the reverse of multiplication. It's the single most useful algebraic skill for solving equations.
Always look for GCF first!
25x² − 49 = (5x + 7)(5x − 7)
For ax² + bx + c, find two numbers that multiply to ac and add to b:
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)
For polynomials with 4+ terms, group pairs and extract common factors:
Group: x²(x + 1) + 2(x + 1) = (x² + 2)(x + 1)
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.
Factor: (x+3)(x−3) / (x+2)(x+3)
Cancel (x+3): (x−3)/(x+2), valid for x ≠ −3, x ≠ −2
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.