Polynomials & Factoring — Core Skills

Polynomials and Factoring: The Basics

Part 1 of 2 — One Skill, One Idea

A polynomial is an expression made of terms added or subtracted together, where each term is a number times a power of $x$. For example, $4x^{3} - 2x^{2} + 7$.

Three words to know:

• The exponent is the small raised number. It tells you how many times to multiply $x$ by itself.
• The degree is the largest exponent in the whole polynomial. In $4x^{3} - 2x^{2} + 7$, the degree is $3$.
• The leading coefficient is the number in front of the term with the largest exponent. Here it is $4$.

To factor means to rewrite something as a multiplication.

The one move: pull out what every term shares

This is called factoring out the greatest common factor, or GCF. Look at every term. Find the biggest number and the most $x$ letters that all of them share. Pull that out in front.

Worked example

Factor $6x^{2} + 15x$ completely.

Step 1 — Look at the numbers, $6$ and $15$. The biggest number that divides both is $3$.

Step 2 — Look at the letters. The first term has $x^{2}$, which is $x \times x$. The second term has one $x$. Both have at least one $x$, so you can pull out one $x$.

Step 3 — The GCF is $3x$. Write it in front of a new set of parentheses: $3x( \quad )$.

Step 4 — Divide each original term by $3x$ to fill the parentheses. First, $6x^{2} \div 3x = 2x$. Second, $15x \div 3x = 5$.

Step 5 — Write the answer: $3x(2x + 5)$.

Check by multiplying back out: $3x \times 2x = 6x^{2}$ and $3x \times 5 = 15x$. That matches, so the factoring is correct.

Zeros and factors

A zero of a function is a value of $x$ that makes the function equal $0$. If $f(3) = 0$, then $3$ is a zero, and that also means $(x - 3)$ is one of the factors of $f(x)$. Zeros and factors are two ways of saying the same thing.

Polynomials and Factoring: Practice

Part 2 of 2 — Run the Steps

Factor in this order every time:

1. GCF first. Pull out the biggest number and the most $x$ letters that every term shares.
2. Check for a difference of squares. If what is left looks like $x^{2} - (\text{a number})$ and that number is a perfect square, it factors as $(x - n)(x + n)$.
3. Otherwise use the two-number trick. For $x^{2} + bx + c$, find two numbers that multiply to $c$ and add to $b$.
4. Multiply back out to check. If you get the original expression, you factored correctly.

The perfect squares worth memorizing

$4, 9, 16, 25, 36, 49, 64, 81, 100$

These come from $2^{2}, 3^{2}, 4^{2}, 5^{2}, 6^{2}, 7^{2}, 8^{2}, 9^{2}, 10^{2}$.

Reading zeros off a factored polynomial

Once a polynomial is written as factors multiplied together, set each factor equal to $0$ and solve. For $f(x) = (x - 1)(x + 8)$, the zeros are $x = 1$ and $x = -8$. Each zero is also a place where the graph touches or crosses the $x$-axis.