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College Algebra Unit 2: Polynomials & factoring

Polynomials are sums of terms like axⁿ. Factoring them is the key skill for solving equations and simplifying rational expressions.

Covers: Operations on polynomials, factoring patterns, rational expressions

Study guide

Polynomials are sums of terms with whole-number exponents. Add and subtract them by combining like terms; multiply by distributing every term, with FOIL as a shortcut for two binomials. Special products include (a ± b)² = a² ± 2ab + b² and (a + b)(a − b) = a² − b².

Factoring writes a polynomial as a product. Always start with the greatest common factor. Then check patterns: difference of squares a² − b² = (a − b)(a + b), perfect-square trinomials, sum and difference of cubes a³ ± b³ = (a ± b)(a² ∓ ab + b²), and trinomials x² + bx + c, which factor as (x + m)(x + n) where mn = c and m + n = b. For ax² + bx + c, use the ac method or trial and error. Four-term polynomials often factor by grouping. A sum of squares a² + b² doesn't factor over the reals.

Rational expressions are fractions of polynomials. Simplify by factoring and canceling common factors, noting the restricted values that make the original denominator zero. Multiply by multiplying numerators and denominators; divide by multiplying by the reciprocal. Add or subtract with a least common denominator, then simplify. Complex fractions (fractions within fractions) simplify by multiplying the top and bottom by the LCD of all the small fractions.

Expressions such as x⁴ − 5x² + 4 factor like quadratics. Let u = x² to get (u − 1)(u − 4) = (x² − 1)(x² − 4), then factor each difference of squares. To solve an equation with variables in denominators, multiply both sides by the LCD, solve the resulting equation, and reject any solution that makes an original denominator zero.

Key ideas

• Add and subtract polynomials by combining like terms; multiply by distributing every term.
• Always factor out the greatest common factor first.
• Know the special patterns: difference of squares, perfect square trinomials, sum and difference of cubes.
• Factor trinomials ax² + bx + c by finding two numbers that multiply to ac and add to b (the ac method).
• Simplify a rational expression by factoring top and bottom and canceling common factors, noting excluded values.

Formulas and rules

Common mistakes

• Canceling terms instead of factors, like crossing out x in (x + 2)/x.
• Forgetting to list values that make the denominator zero.

Words to know

Difference of squares

a² − b² = (a − b)(a + b)

Difference of cubes

a³ − b³ = (a − b)(a² + ab + b²)

Greatest common factor (GCF)

The largest factor shared by every term; pull it out first.

Factoring x² + bx + c

Find two numbers that multiply to c and add to b.

Restricted values

x-values that make a denominator 0; excluded even after canceling.

Sum of squares

a² + b² doesn't factor over the real numbers.

Polynomial

A sum of terms with whole-number exponents.

Sum of cubes

a³ + b³ = (a + b)(a² − ab + b²)

Perfect-square trinomial

a² ± 2ab + b² = (a ± b)²

Factoring by grouping

Pair terms, factor each pair, pull out the common binomial.

Rational expression

A fraction of polynomials.

Complex fraction

A fraction within a fraction; multiply by the LCD of all small fractions.

Sample practice questions

31 more questions like these, from level 1 to 5, are in the practice set.