## Introduction

A **polynomial** is an expression created by the **sum of mathematical terms**. Understanding polynomial expressions can help you understand **graphs of lines**, **parabolas**, and other **functions**.

## Polynomials

The terms of a polynomial have specific characteristics. See this list to determine what is a term and what is not.

**Examples of Terms:**

- $3$
- $x$
- $3x^{2}$
- $\frac{1}{2}x$
- $x^{4}$

**Example of Not Terms:**

- $x^{-2}$
- $\frac{1}{x^{4}}$
- $\sqrt{16}$

A polynomial is made up of **more than one term**, and the **standard form** is written by listing the **exponents in decreasing order**. Here is an example of a polynomial written in the standard form. The labels indicate the numbers that are the **coefficients** and the **constant**. This polynomial has a degree of 3, the highest exponent shown, and it is a trinomial because there are three terms: $6x^{3} +4x^{2} +10$.

For terms with more than one variable, add the exponents together to determine the degree: $4x^{2}y^{2}$.

This polynomial has a degree of 7 and is a binomial because there are 2 terms: $2x^{2}y^{5} + x^{3}$.

## Polynomial Operations

Just like numbers, polynomials can be added, subtracted, multiplied, and divided. Here are some examples.

### Adding and Subtracting Polynomials

**To add or subtract polynomials, combine all like terms**: $(2x^{2} + 3) + (x^{2} - 10) = 3x^{2} + -7$

### Multiplying Polynomials

**To multiply polynomials, use the Distributive Property**: $(3x)\times (2x^{2} +4x+3)=6x^{3}+12x^{2}+9x$

This problem calculates the product of two binomials, a special case. To multiply, use the **mnemonic device FOIL** (**First - Outer - Inner - Last**).

- To foil, first multiply the
**first two terms**of each binomial; - Next, multiply the
**outer two terms**; - Next, multiply the
**inner two terms**; - And last, multiply the
**last two terms**:

$(x+2)\times (x+3)=x^{2}+3x+2x+6=x^{2}+5x+6$

### Dividing Polynomials

**Dividing polynomials** is similar to dividing fractions. Like division of numbers, you subtract multiples of the divisor from the dividend as you work through the problem, and it is possible to have a **remainder**.

Shown here, the operation of division undoes the product from the previous example. The **divisor** and the **quotient** are the same as the above binomial factors:

$\begin{align} x+3\\ x+2 ~\overline{\big)x^2+5x+6}\\ \underline{-(x^2 +2x)}\\ 3x+6\\ \underline{-3x+6}\\ 0 \end{align}$

## Factoring Polynomials

**To factor, set polynomials equal to zero.** The factoring of polynomials reveals the roots or zeros of an equation and determines **where the graph touches the x-axis at point (x , 0)**.

### Factoring out the GCF

Look for factors common to each of the terms in a polynomial then **factor the common factor** out of the expression**.

This expression has a common factor of 3x, so by **reversing the Distributive Property** we move the term outside parentheses and leave what’s left of the expression inside the parentheses. Although this expression is not fully factored, factoring out the **GCF** makes it easier to complete the factoring process.

$\begin{align} 6x^{3} + 3x^{2} +9x &= 0 \\ 3x(2x^{2} +x +3) &= 0 \end{align}$

## Factoring Trinomials

**To factor trinomials**, there are several different strategies available.

### Factoring with Reverse FOIL: Trinomials with $a = 1$

This trinomial is in the **standard quadratic form** $ax^{2} + bx + c = 0$.

For trinomials with a is equal to 1, factor the trinomial by **reversing the FOIL method**. A shortcut is to determine the factors of c that will sum to b.

$\begin{align} ax^{2} + bx + c &= 0\\ x^{2} +9x +20&=0\\ (x+4)(x+5) &= 0 \end{align}$

### Factoring by Grouping: Trinomials with $a\neq 1$

This trinomial is also in the **standard quadratic form** $ax^{2} + bx + c = 0$, but a is not equal to 1, so the factoring process is different. Multiply a and c together then determine the factors of that product that will sum to b. Next **factor by grouping**.

$\begin{align} ax^{2} + bx + c &= 0\\ 2x^{2} +x - 6 &= 0\\ 2x^{2} -3x +4x - 6 &= 0\\ \left(2x^{2}-3x\right) + \left(4x - 6\right) &= 0\\ x(2x -3) +2(2x -3)&=0\\ (x+2)(2x-3) &= 0 \end{align}$

### Factoring by Grouping: Polynomials with Higher Degrees

You can also **factor polynomials by grouping** when the **degree is higher than 2**.

This polynomial has a degree of 3 and 4 terms.

$\begin{align} x^{3} +2x^{2} -4x -8 &= 0\\ (x^{3} +2x^{2}) + (-4x -8)&=0\\ x^{2}(x+2) + -4(x+2) &= 0\\ (x^{2} -4)(x+6)&=0 \end{align}$