Polynomials and Factoring

What is a polynomial?

A monomial is a number, a variable, or a product of a number and variables with whole-number exponents: 7, x, −3x² and 4xy are monomials. A polynomial is a monomial or a sum of monomials, which are called its terms. The number in front of a term is its coefficient.

The degree of a polynomial in x is its highest power of x. It is in standard form when the terms go from the highest power down to the lowest. The first coefficient in standard form is the leading coefficient. In −2x³ + 5x − 1, the degree is 3 and the leading coefficient is −2.

Expressions like 1/x or √x are not polynomials, because their exponents are not whole numbers (1/x = x⁻¹ and √x = x^(1/2)).

Your turn

Write 3x − 7 + x³ in standard form. Give its degree and leading coefficient.

Adding and subtracting polynomials

Like terms have the same variable raised to the same power: 3x² and −5x² are like terms, but 3x² and 3x are not. To add polynomials, combine like terms by adding their coefficients. To subtract, change the sign of every term in the polynomial being subtracted, then add.

Multiplying polynomials

When you multiply powers of the same variable, add the exponents: x² · x³ = x⁵. Multiply the coefficients as usual: 4x · 3x² = 12x³.

A monomial times a polynomial: distribute

Two binomials: FOIL

Every term in the first bracket multiplies every term in the second. For two binomials that is four products, often remembered as FOIL: First, Outer, Inner, Last.

The area model (box method)

Draw a box with one factor’s terms across the top and the other’s down the side. Each cell is a product. Negative terms work too: the box isn’t a real rectangle any more, but the bookkeeping is the same.

FOIL only names the four products of two binomials. With longer polynomials, keep going: every term times every term.

Special products

Three products come up so often that they are worth knowing by heart.

In (a + b)², the 2ab comes from the two middle products ab and ba: in a square like (x + 3)², they are two matching strips. In (a + b)(a − b), the middle products are −ab and ab, and they cancel.

Factoring out the greatest common factor

Factoring means writing a polynomial as a product. Always start by looking for the greatest common factor (GCF): the largest number, times the highest power of x, that divides every term. Then divide each term by it.

Check by multiplying back: 6x²(2x − 3) = 12x³ − 18x² ✓.

Factoring trinomials

When the leading coefficient is 1

Since (x + p)(x + q) = x² + (p + q)x + pq, to factor x² + bx + c find two numbers that multiply to c and add to b.

When the leading coefficient isn’t 1: the ac method

For ax² + bx + c, find two numbers that multiply to a · c and add to b. Use them to split the middle term, then factor the two pairs (this is factoring by grouping).

If no pair of whole numbers works, the trinomial doesn’t factor over the integers. For example, nothing multiplies to 1 and adds to 1, so x² + x + 1 stays as it is.

Special factoring patterns

Read the special products backwards.

Difference of squares

a² − b² = (a + b)(a − b)

A sum of squares such as x² + 25 does not factor over the real numbers.

Perfect-square trinomials

a² + 2ab + b² = (a + b)² and a² − 2ab + b² = (a − b)²

Factoring completely

A polynomial is factored completely when no bracket can be factored any further. A good order of checks:

1. Take out the GCF.
2. Count the terms. Two terms: is it a difference of squares (or of cubes)? Three terms: is it a perfect square, or does the a = 1 or ac method work?
3. Check every bracket again.
4. Multiply back to check.

A taste of cubes

Sums and differences of cubes also factor, into a binomial times a trinomial:

a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)

The trinomial x² + 2x + 4 does not factor over the integers. You will meet cubes again in later courses; for now, check the pattern by multiplying it out.

Common mistakes

Practice problems

Printable worksheet (PDF)

Work each one on paper, then open the answer to check your work. Problem 5 has a surprise.

1. Simplify: (4x² − 3x + 2) − (x² + 5x − 7)
   Show answer

   4x² − 3x + 2 − x² − 5x + 7Change every sign in the second bracket3x² − 8x + 9Combine like terms
2. Multiply: −2x(3x² − x + 5)
   Show answer

   −2x · 3x² − 2x · (−x) − 2x · 5Distribute −2x−6x³ + 2x² − 10xMultiply each term
3. Multiply: (3x − 2)(x + 5)
   Show answer

   3x² + 15x − 2x − 10FOIL3x² + 13x − 10Combine like terms
4. Multiply: (x − 6)²
   Show answer

   x² − 2 · x · 6 + 6²(a − b)²x² − 12x + 36Not x² + 36
5. Factor out the GCF: 15x³ + 10x² − 5x
   Show answer

   5x(3x² + 2x − 1)The GCF is 5x; −5x ÷ 5x = −15x(3x − 1)(x + 1)The trinomial factors too, so this is factored completely
6. Factor: x² + 9x + 20
   Show answer

   4 · 5 = 20 and 4 + 5 = 9Multiply to 20, add to 9(x + 4)(x + 5)
7. Factor: x² + 2x − 24
   Show answer

   6 · (−4) = −24 and 6 + (−4) = 2Multiply to −24, add to 2(x + 6)(x − 4)
8. Factor: 5x² − 13x − 6
   Show answer

   5 · (−6) = −30a · c−15 · 2 = −30 and −15 + 2 = −13Multiply to −30, add to −135x² − 15x + 2x − 6Split the middle term5x(x − 3) + 2(x − 3)Factor each pair(x − 3)(5x + 2)Factor out (x − 3)
9. Factor: 49x² − 1
   Show answer

   (7x)² − 1²Difference of squares(7x + 1)(7x − 1)
10. Factor completely: 3x³ − 12x² + 12x
    Show answer

    3x(x² − 4x + 4)Take out the GCF, 3x3x(x − 2)²A perfect-square trinomial

History: from pictures to letters

Long before algebra had symbols, Greek geometers worked with areas. Book II of Euclid’s Elements states, in Proposition 4, that if a straight line is cut at random, the square on the whole line equals the squares on the two pieces plus twice the rectangle made by the pieces. Draw it and you get the area model at the top of this page; in today’s symbols it says (a + b)² = a² + 2ab + b². Euclid wrote it about lines and areas, not letters and numbers.

Writing such rules with letters came much later. In 1591 the French mathematician François Viète published In artem analyticam isagoge (Introduction to the analytic art) at Tours. MacTutor describes it as the first systematic algebraic notation: Viète used vowels for unknown quantities and consonants for known ones, and he is credited with the word coefficient.

The habit of using letters from the end of the alphabet, like x, for unknowns and letters from the start, like a, b and c, for known numbers comes from René Descartes. His La Géométrie, published in 1637 at Leiden as an appendix to his Discours de la méthode, used that convention, and it is still how this page writes ax² + bx + c.

Connections to quadratic equations

In the Quadratic Equations unit, factoring was one way to solve an equation. Factor the quadratic, then use the zero-product property: if (x − 5)(x + 3) = 0, then x = 5 or x = −3. The factors and the roots match up: a factor (x − r) gives the root x = r, which is where the parabola crosses the x-axis.

The special products are behind completing the square, too: x² + 6x + 9 = (x + 3)² is a perfect-square trinomial. And if a quadratic doesn’t factor over the integers, like x² + x − 1, the quadratic formula still finds its roots.

Sources

• Euclid, Elements, Book II, Proposition 4, in D. E. Joyce’s online edition, Clark University: Proposition II.4 (the square on a whole line equals the squares on the two segments plus twice the rectangle contained by them).
• J. J. O’Connor and E. F. Robertson, MacTutor History of Mathematics, University of St Andrews: François Viète (In artem analyticam isagoge, Tours, 1591; vowels for unknowns and consonants for knowns; the word “coefficient”; Descartes’s later convention of letters at the end of the alphabet for unknowns).
• J. J. O’Connor and E. F. Robertson, MacTutor: René Descartes (La Géométrie, published at Leiden in 1637 as an appendix to the Discours de la méthode).
• Common Core State Standards for Mathematics: Arithmetic with Polynomials and Rational Expressions (A-APR.A.1, adding, subtracting and multiplying polynomials) and Seeing Structure in Expressions (A-SSE.A.2 and A-SSE.B.3, rewriting and factoring), which this lesson supports.

Credits: all diagrams on this page are original SVG drawings made for Jugami. Every product, factorization and step was checked by a script.