Example 3.1.
Evaluating \(4x^2-2x+1\) at \(x=-1\) results in
\begin{equation*}
4(-1)^2-2(-1)+1 = 4+2+1 =7.
\end{equation*}
Section 3.1 What Are Expressions?
An algebraic expression is a quantity representing a real number which may include one or more variables holding place for real numbers with unspecified values. The following are algebraic expressions:
\begin{equation*}
4x^2-2x+1,\quad 10^{x^2-y^2}, \quad \frac{9}{\sqrt{4-\theta^2}}.
\end{equation*}
We may evaluate algebraic expressions by replacing each variable with a chosen real number.
We often need to simplify algebraic expressions. What this means depends on the situation, but usually we perform all operations we can to obtain a reduced expression.
Example 3.2. Expanding an Expression.
Simplifying
\begin{equation*}
x (3y - 5z) + 2xz
\end{equation*}
might require expanding the product via distribution and combining any like terms. Let’s do this, separating equal expressions with the equal sign and writing down the page so we can compare each step with the previous:
\begin{align*}
x (3y - 5z) + 2xz \quad
\amp= x(3y) - x(5z) +2xz\\
\amp= 3xy - 5xz + 2xz\\
\amp= 3xy + (2-5)xz\\
\amp= 3xy -3xz.
\end{align*}
Remark 3.3. Mathematical Exposition.
Evaluating or simplifying an expression requires writing a sequence of equal expressions separated by the equal sign “\(=\)”. This is a practice you should adhere to throughout your college mathematics courses.
