###### In this section, you will...

review definitions surrounding integers.

add, subtract, multiply, divide, and reduce fractions

use the order of operations to put expressions into simplest form

do computations with percents

review radicals

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Algebra is often described as the generalization of arithmetic. The systematic use of variables, letters used to represent numbers, allows us to communicate and solve a wide variety of real-world problems. For this reason, we begin by reviewing real numbers and their operations.

review definitions surrounding integers.

add, subtract, multiply, divide, and reduce fractions

use the order of operations to put expressions into simplest form

do computations with percents

review radicals

In algebra, letters called variables are used to represent numbers. Combinations of variables and numbers along with mathematical operations form algebraic expressions, or just expressions. The following are some examples of expressions with one variable, \(x\text{:}\)

\begin{gather*}
2x+3,~~~ x^2-9,~~~ \frac{1}{x}+\frac{x}{x+2}, ~~~ 3\sqrt{x}+x
\end{gather*}

Note that to be an expression, the numbers, variables, and operations must be in an order that "makes sense". In other words, they need to be in an order so that the expression could be evaluated if the variables were given numeric values. For example, it doesn't make sense to have \(\div2^+\text{,}\) so this is not an expression.

Terms in an algebraic expression are separated by addition operators and factors are separated by multiplication operators. The numerical factor of a term is called the coefficient. For example, the algebraic expression \(x^2y^2+6xy-3\) can be thought of as \(x^2y^2+6xy+(-3)\) and has three terms. The first term, \(x^2y^2\text{,}\) represents the quantity \(1x^2y^2=1\cdot x\cdot x\cdot y\cdot y\) where \(1\) is the coefficient and \(x\) and \(y\) are the variables. All of the variable factors with their exponents form the variable part of a term. If a term is written without a variable factor, then it is called a constant term. Below we see the components of \(x^2y^2+6xy-3\text{.}\)

Terms | Coefficient | Variable Part |

\(x^2y^2\) | \(1\) | \(x^2y^2\) |

\(6xy\) | \(6\) | \(xy\) |

\(-3\) | \(-3\) |

Notice that the first term has a coefficient of \(1\text{.}\) Since we can always multiply an expression by \(1\) without changing its value, we say that a term without a coefficient has a coefficient of \(1\text{.}\)

The third term in this expression, \(-3\text{,}\) is called a constant term because it is written without a variable factor. While a variable represents an unknown quantity and may change, the constant term does not change. Also, notice that the constant term is \(-3\text{,}\) not just \(3\text{.}\) Since terms are defined by addition, whenever we see subtraction, we can think of it as making the coefficient negative.

List all coefficients and variable parts of each term in the expression \(10a^2-5ab-b^2\text{.}\)

Solution

We want to think of the third term in this example, \(-b^2\text{,}\) as \(-1b^2.\)

Terms | Coefficient | Variable Part |

\(10a^2\) | \(10\) | \(a^2\) |

\(-5ab\) | \(-5\) | \(ab\) |

\(-b^2\) | \(-1\) | \(b^2\) |

In our study of algebra, we will encounter a wide variety of algebraic expressions. Typically, expressions use the two most common variables, \(x\) and \(y\text{.}\) However, expressions may use any letter (or symbol) for a variable, even Greek letters, such as alpha (\(\alpha\)) and beta (\(\beta\)). Some letters and symbols are reserved for special constants, such as \(\pi\approx 3.14159\text{.}\)

The properties of real numbers are important in our study of algebra because a variable is simply a letter that represents a real number. In particular, the distributive property states that if given any real numbers \(a\text{,}\) \(b\) and \(c\text{,}\) then

\begin{equation*}
a(b+c)=ab+ac.
\end{equation*}

Since the order does not matter for multiplication, we can apply the distributive property on the other side as well:

\begin{equation*}
(b+c)a=ba+ca
\end{equation*}

This property is one that we apply often when simplifying algebraic expressions. To demonstrate how it will be used, we simplify \(2(5-3)\) in two ways, and observe the same correct result.

\begin{gather*}
\text{Working parenthesis first: } 2(5-3)=2(2)=4,\\
\text{Using the distributive property: } 2(5-3)=2\cdot 5-2\cdot 3=10-6=4.
\end{gather*}

Certainly, if the contents of the parentheses can be simplified we should do that first. On the other hand, when the contents of parentheses cannot be simplified any further, we multiply every term within it by the factor outside of it using the distributive property. Applying the distributive property allows us to multiply and remove the parentheses.

Simplify \(5(-2a+5b)-2c\text{.}\)

Solution

Multiply only the terms grouped within the parentheses for which we are applying the distributive property.

\begin{equation*}
\begin{aligned}
5(-2a+5b)-2c \amp = 5\cdot (-2a)+5\cdot 5b-2c\\
\amp = -10a+25b-2c.
\end{aligned}
\end{equation*}

Terms whose variable parts have the same variables with the same exponents are called like terms. Furthermore, constant terms are considered to be like terms. If an algebraic expression contains like terms, we can apply the distributive property as follows:

\begin{equation*}
\begin{aligned}
5x+7x \amp = (5+7)x = 12x\\
4x^2+5x^2-7x^2\amp=(4+5-7)x^2=2x^2.
\end{aligned}
\end{equation*}

In other words, if the variable parts of terms are exactly the same, then we can add or subtract the coefficients to obtain the coefficient of a single term with the same variable part. This process is called combining like terms. For example,

\begin{equation*}
12x^2y^3+3x^2y^3=15x^2y^3.
\end{equation*}

Notice that the variable factors and their exponents do not change. Combining like terms in this manner, so that the expression contains no other similar terms, is called simplifying the expression. Use this idea to simplify algebraic expressions with multiple like terms.

Simplify \(x^2-10x+8+5x^2-6x-1\text{.}\)

Solution

Identify the like terms and add the corresponding coefficients.

\begin{equation*}
\begin{aligned}
1x^2-10x+8+5x^2-6x-1
= 6x^2-16x+7
\end{aligned}
\end{equation*}

Simplify \(a^2b^2-ab-2(2a^2b^2-7ab+1+2ab)\text{.}\)

Solution

Before applying the distributive property, we should try to combine like terms.

\begin{equation*}
\begin{aligned}
a^2b^2-ab-2(2a^2b^2-7ab+1+2ab)\amp=a^2b^2-ab-2(2a^2b^2-5ab+1)
\end{aligned}
\end{equation*}

Next we distribute \(-2\) and then combine like terms.

\begin{equation*}
\begin{aligned}
a^2b^2-ab-2(2a^2b^2-5ab+1) \amp = a^2b^2-ab-4a^2b^2+10ab-2\\
\amp = -3a^2b^2+9ab-2
\end{aligned}
\end{equation*}

An algebraic expression can be thought of as a generalization of particular arithmetic operations. Performing these operations after substituting given values for variables is called evaluating. In algebra, a variable represents an unknown value. However, if the problem specifically assigns a value to a variable, then you can replace that letter with the given number and evaluate using the order of operations.

Evaluate:

- \(5x-2\) where \(x=\frac{2}{3}\text{,}\)
- \(y^2-y-6\) where \(y=-4\text{.}\)

Solution

We replace, or substitute, the given value for the variable.

- \(\begin{aligned}5x-2 \amp= 5 \left(\frac{2}{3}\right)-2\\ \amp =\frac{10}{3}-\frac{2}{1}\cdot\frac{3}{3}\\ \amp=\frac{10-6}{3}\\ \amp=\frac{4}{3}, \end{aligned}\)
- \(\begin{aligned}y^2-y-6 \amp= (-4)^2-(-4)-6\\ \amp =16+4-6\\ \amp=14. \end{aligned}\)

Evaluate:

- \(1+x+x^2+x^3\) where \(x=1\text{,}\)
- \(1+x+x^2+x^3\) where \(x=-1\text{.}\)

Solution

We replace, or substitute, the given value for the variable.

- \(\begin{aligned}1+x+x^2+x^3 \amp= 1+(1)+(1)^2+(1)^3 \\ \amp=1+1+1+1 \\ \amp=4 \end{aligned}\)
- \(\begin{aligned}1+x+x^2+x^3 \amp= 1+(-1)+(-1)^2+(-1)^3 \\ \amp=1-1+1-1 \\ \amp=0 \end{aligned}\)

Often algebraic expressions will involve more than one variable.

Evaluate \(a^3-8b^3\) where \(a=-1\) and \(b=\frac{1}{2}\text{.}\)

Solution

After substituting in the appropriate values, we must take care to simplify using the correct order of operations.

\begin{equation*}
\begin{aligned}
a^3-8b^3 \amp = (\alert{-1})^3-8\left(\alert{\frac{1}{2}}\right)^3\\
\amp = -1-8\left(\frac{1}{8}\right)\\
\amp = -1-1\\
\amp = -2
\end{aligned}
\end{equation*}

Evaluate \(\frac{x^2-y^2}{2x-1}\) where \(x=-\frac{3}{2}\) and \(y=-3\text{.}\)

Solution

\begin{equation*}
\begin{aligned}
\frac{x^2-y^2}{2x-1} \amp = \frac{\left(\alert{-\frac{3}{2}}\right)^2-(\alert{-3})^2}{2\left(\alert{-\frac{3}{2}}\right)-1}\\
\amp = \frac{\frac{9}{4}-9}{-3-1}
\end{aligned}
\end{equation*}

At this point we have a complex fraction. Simplify the numerator and then multiply by the reciprocal of the denominator.

\begin{equation*}
\begin{aligned}
\amp = \frac{\frac{9}{4}-\frac{9}{1}\cdot\alert{\frac{4}{4}}}{-4}\\
\amp = \frac{-\frac{27}{4}}{\frac{-4}{1}}\\
\amp = \frac{-27}{4}\left(-\frac{1}{4}\right)\\
\amp=\frac{27}{16}
\end{aligned}
\end{equation*}

The answer to the previous example can be written as a mixed number, \(\frac{27}{16}=1\frac{11}{16}\text{.}\) Unless the original problem has mixed numbers in it, or it is an answer to a real-world application, solutions in this course will be expressed as reduced improper fractions.

Evaluate \(\sqrt{b^2-4ac}\) where \(a=-1,\, b=-7\) and \(c=\frac{1}{4}\text{.}\)

Solution

Substitute in the appropriate values and then simplify. \(\begin{aligned} \sqrt{b^2-4ac} \amp = \sqrt{(\alert{-7})^2-4(\alert{-1})\left(\alert{\frac{1}{4}}\right)}\\ \amp = \sqrt{49+4\left(\frac{1}{4}\right)}\\ \amp = \sqrt{49+1}\\ \amp = \sqrt{50}\\ \amp = \sqrt{25\cdot 2}\\ \amp = 5\sqrt{2} \end{aligned}\)