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Algebra 1 Unit 6: Polynomials & factoring

Polynomials are sums of terms with whole-number exponents. Factoring rewrites them as products, which is the main tool for solving quadratics.

Covers: Adding and multiplying polynomials, GCF, factoring trinomials, difference of squares

Study guide

A polynomial is a sum of terms like 4x³ − x + 2. Its degree is the highest exponent. To add or subtract polynomials, combine like terms, distributing a negative sign through every term when subtracting. To multiply, distribute every term in one polynomial to every term in the other; for two binomials, FOIL (first, outer, inner, last) is a shortcut.

Factoring reverses multiplication. Always start by pulling out the greatest common factor (GCF), the largest factor every term shares: 3x² + 6x = 3x(x + 2).

To factor a trinomial x² + bx + c, find two numbers that multiply to c and add to b: x² + 7x + 10 = (x + 2)(x + 5). When the leading coefficient isn't 1 (ax² + bx + c), find two numbers that multiply to ac and add to b, split the middle term, and factor by grouping.

Special patterns save time. A difference of squares factors as a² − b² = (a − b)(a + b), so x² − 16 = (x − 4)(x + 4). A perfect-square trinomial factors as a² ± 2ab + b² = (a ± b)². A sum of squares like x² + 9 does not factor over the real numbers. Check any factoring by multiplying back out.

A polynomial is in standard form when its terms are written from highest degree to lowest. The leading coefficient is the coefficient of that first, highest-degree term, and the constant term is the term with no variable. A monomial has one term, a binomial two and a trinomial three. By degree, a polynomial is constant (0), linear (1), quadratic (2) or cubic (3).

Factoring completely means continuing until no factor can be factored further. For example, 2x³ − 18x = 2x(x² − 9) = 2x(x − 3)(x + 3).

Key ideas

• Add polynomials by combining like terms.
• Multiply by distributing every term (FOIL for two binomials).
• Always look for a greatest common factor first.
• x² + bx + c: find two numbers that multiply to c and add to b.
• Difference of squares: a² − b² = (a − b)(a + b).

Formulas and rules

Common mistakes

• Sign errors when factoring trinomials with negative terms.
• Forgetting to factor out a GCF first.

Words to know

FOIL

First, Outer, Inner, Last: multiplying two binomials.

Factoring x² + bx + c

Find two numbers that multiply to c and add to b: x² + 7x + 10 = (x + 2)(x + 5).

Difference of squares

a² − b² = (a − b)(a + b)

Greatest common factor (GCF)

The largest factor shared by every term: 3x² + 6x = 3x(x + 2).

Degree of a polynomial

The highest exponent (4x³ − x + 2 has degree 3).

Polynomial

A sum of terms with whole-number exponents, like 4x³ − x + 2.

Monomial / binomial / trinomial

Polynomials with one, two and three terms.

Standard form (polynomial)

Terms written from highest to lowest degree.

Factoring by grouping

Group terms in pairs, factor each pair, then factor out the common binomial.

Perfect-square trinomial

a² ± 2ab + b² = (a ± b)²

Factoring ax² + bx + c

Find two numbers that multiply to ac and add to b, then group.

Prime polynomial

A polynomial that can't be factored, like x² + 9 over the reals.

Sample practice questions

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