# 4.9: Summary of Key Concepts

- Page ID
- 58531

**Algebraic Expressions **

An **algebraic expression** (often called simply an expression) is a number, a letter, or a collection of numbers and letters along with meaningful signs of operation. (5÷0 is not meaningful.)

**Terms **

In an algebraic expression, the quantities joined by "+" signs are **terms**.

**Distinction Between Terms and Factors**

**Terms** are parts of sums and are therefore separated by addition signs. **Factors** are parts of products and are therefore separated by multiplication signs.

**Common Factors**

In an algebraic expression, a factor that appears in **every** term, that is, a factor that is common to each term, is called a **common factor**.

**Coefficients **

The **coefficient** of a quantity records how many of that quantity there are. The coefficient of a group of factors is the remaining group of factors.

**Distinction Between Coefficients and Exponents **

**Coefficients** record the number of like terms in an expression.

\(\underbrace{x+x+x}_{3 \text { terms }}=\begin{array}{c}

3 x \\

\text { coefficient is } 3

\end{array}\)

**Exponents** record the number of like factors in an expression

\(\underbrace{x \cdot x \cdot x}_{3 \text { factors }}=\begin{array}{c}

x^{3} \\

\text { exponent is } 3

\end{array}\)

**Equation **

An **equation** is a statement that two expressions are equal.

**Numerical Evaluation**

**Numerical evaluation** is the process of determining a value by substituting numbers for letters.

**Polynomials **

A polynomial is an algebraic expression that does not contain variables in the denominators of fractions and in which all exponents on variable quantities are whole numbers.

A **monomial** is a polynomial consisting of only one term.

A **binomial** is a polynomial consisting of two terms.

A **trinomial** is a polynomial consisting of three terms.

**Degree of a Polynomial**

The degree of a term containing one variable is the value of the exponent on the variable.

The degree of a term containing more than one variable is the sum of the exponents on the variables.

The degree of a polynomial is the degree of the term of the highest degree.

**Linear Quadratic Cubic Polynomials **

Polynomials of the first degree are **linear** polynomials.

Polynomials of the second degree are **quadratic** polynomials.

Polynomials of the third degree are **cubic** polynomials.

**Like Terms **

**Like terms** are terms in which the variable parts, including the exponents, are identical.

**Descending Order**

By convention, and when possible, the terms of an expression are placed in descending order with the highest degree term appearing first. \(5x^3−2x^2+10x−15\) is in descending order.

**Multiplying a Polynomial by a Monomial **

To multiply a polynomial by a monomial, multiply every term of the polynomial by the monomial and then add the resulting products together.

\(7(x−3)=7x−7 \cdot 3=7x−21\)

**Simplifying **\(+(a+b)\) **and **\(−(a+b)\)

\(\begin{array}{l}

+(a+b)=a+b \\

-(a+b)=-a-b

\end{array}\)

**Multiplying a Polynomial by a Polynomial**

To multiply polynomials together, multiply every term of one polynomial by every term of the other polynomial.

\(\begin{aligned}

(x+3)(x-4) &=x^{2}-4 x+3 x-12 \\

&=x^{2}-x-12

\end{aligned}\)

**Special Products**

\(\begin{aligned}

(a+b)^{2} &=a^{2}+2 a b+b^{2} & \text { Note }: &(a+b)^{2} \neq a^{2}+b^{2} \\

(a-b)^{2} &=a^{2}-2 a b+b^{2} &(a-b)^{2} \neq a^{2}-b^{2} \\

(a+b)(a-b) &=a^{2}-b^{2} & &

\end{aligned}\)

**Independent and Dependent Variables **

In an equation, any variable whose value can be freely assigned is said to be an **independent variable**. Any variable whose value is determined once the other values have been assigned is said to be a **dependent variable**.

**Domain **

The collection of numbers that can be used as replacements for the independent variable in an expression or equation and yield a meaningful result is called the **domain** of the expression or equation.