Polynomials & Factoring

1. Introduction

A polynomial is one of the most important objects in all of algebra: an expression built from a variable using only addition, subtraction, and multiplication by constants and whole-number powers. Examples include $3x^2 - 5x + 2$, $x^4 - 1$, and even a plain number like $7$. Polynomials model curves, projectile paths, costs, areas, and countless physical quantities, and they are the bridge between elementary algebra and the study of functions and calculus.

Factoring is the reverse of multiplying: it rewrites a polynomial as a product of simpler polynomials. Why bother? Because factoring is the key that unlocks solving. By the Zero Product Property, a product is zero exactly when one of its factors is zero, so once a polynomial equation is factored we can split it into easy pieces. The values that make a polynomial zero are its roots (or zeros), and they are precisely the $x$-intercepts of its graph.

This article covers what polynomials are and how to add, subtract, and multiply them; the standard factoring patterns (GCF, trinomials, difference of squares, perfect squares, sum and difference of cubes, and grouping); the Factor Theorem linking roots and factors; and how to use factoring to find roots. We build each technique with derivations, geometric interpretations, and worked examples so the patterns stick permanently.

2. Core Concepts

2.1 Anatomy of a polynomial

A polynomial in $x$ is a sum of terms of the form $a x^n$, where the coefficient $a$ is a real number and the exponent $n$ is a non-negative integer. Key vocabulary:

• Degree: the highest exponent present. In $4x^3 - x + 9$ the degree is $3$.
• Leading coefficient: the coefficient of the highest-degree term (here, $4$).
• Constant term: the term with no variable (here, $9$).
• Standard form: terms written in descending order of degree.

Degree $1$ polynomials are linear, degree $2$ are quadratic, degree $3$ are cubic, and so on. A polynomial of degree $n$ has at most $n$ real roots.

2.2 Adding and subtracting polynomials

To add or subtract polynomials, combine like terms — terms with the same variable and exponent. For instance, $3x^2$ and $-5x^2$ combine to $-2x^2$, but $3x^2$ and $3x$ do not. When subtracting, distribute the negative sign to every term inside the parentheses:

$(2x^2 + 3x - 1) - (x^2 - 5x + 4) = 2x^2 + 3x - 1 - x^2 + 5x - 4 = x^2 + 8x - 5.$

2.3 Multiplying polynomials and FOIL

Multiplication uses the distributive property repeatedly: every term of one factor multiplies every term of the other. For two binomials, the mnemonic FOIL (First, Outer, Inner, Last) organizes the four products:

$(x + 3)(x + 4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12.$

For larger products, distribute systematically. The general pattern for $(x + p)(x + q)$ is:

$(x + p)(x + q) = x^2 + (p + q)x + pq.$

The middle coefficient is the sum $p + q$ and the constant is the product $pq$ — the engine behind trinomial factoring.

2.4 The Zero Product Property and roots

The reason factoring matters so much is the Zero Product Property:

$pq = 0 \quad \Longrightarrow \quad p = 0 \ \text{ or } \ q = 0.$

If we factor a polynomial equation so that a product equals zero, each factor gives a separate, simple equation. The solutions are the roots. Graphically, roots are the $x$-coordinates where the polynomial's graph crosses the $x$-axis.

2.5 The Factor Theorem

There is a precise link between roots and factors called the Factor Theorem: $(x - r)$ is a factor of a polynomial $P(x)$ if and only if $r$ is a root (i.e., $P(r) = 0$). This means a fully factored polynomial displays all its roots at a glance, and conversely, knowing a root immediately hands you a factor. For example, if $P(3) = 0$, then $(x - 3)$ divides $P(x)$.

2.6 Difference of squares (derivation)

The pattern $a^2 - b^2 = (a - b)(a + b)$ comes from expanding the right side:

$(a - b)(a + b) = a^2 + ab - ab - b^2 = a^2 - b^2.$

Recognizing two perfect squares separated by a minus sign is the key. Examples: $x^2 - 9 = (x - 3)(x + 3)$ and $4x^2 - 25 = (2x - 5)(2x + 5)$.

2.7 Perfect-square trinomials

A trinomial is a perfect square when the first and last terms are squares and the middle term is twice their product:

$a^2 + 2ab + b^2 = (a + b)^2, \qquad a^2 - 2ab + b^2 = (a - b)^2.$

For example, $x^2 + 6x + 9 = (x + 3)^2$ because $6 = 2(1)(3)$, and $4x^2 - 12x + 9 = (2x - 3)^2$.

2.8 Sum and difference of cubes

The cube patterns are less intuitive but essential:

$a^3 + b^3 = (a + b)(a^2 - ab + b^2),$ $a^3 - b^3 = (a - b)(a^2 + ab + b^2).$

Notice the sign in the binomial matches the sign between the cubes, while the trinomial's middle sign is opposite. For $x^3 - 8 = x^3 - 2^3$, we get $(x - 2)(x^2 + 2x + 4)$.

2.9 Why $a^2 + b^2$ does not factor over the reals

A frequent temptation is to factor a sum of squares like $x^2 + 9$. Over the real numbers this is irreducible — there is no pair of real binomials whose product is $x^2 + 9$. If $(x + p)(x + q) = x^2 + 9$, then $pq = 9$ and $p + q = 0$, giving $p = -q$ and $p^2 = -9$, impossible for real $p$. Over the complex numbers, $x^2 + 9 = (x - 3i)(x + 3i)$.

2.10 Factoring by grouping

For four-term polynomials, group into two pairs, factor the GCF from each pair, then factor out the common binomial. This works when the polynomial can be written as $ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)$.

2.11 The AC method for $a \neq 1$

When the leading coefficient is not $1$, compute $ac$ and find two numbers multiplying to $ac$ and adding to $b$. Split the middle term and factor by grouping. This generalizes the sum-product pattern from FOIL.

2.12 Polynomial long division (preview)

Dividing one polynomial by another is analogous to long division of integers. When $(x - r)$ divides $P(x)$ evenly, the quotient is a polynomial of degree one less. This connects the Factor Theorem to finding additional factors after one root is known — a technique used heavily in Algebra 2 and beyond.