InterMath Dictionary
Expressions
Parts of an Expression
Examples
TERM
COEFFICIENT
VARIABLE
3x
3
x
1
s
2
3@
@
x and y
2+1=3
Examples of Like Terms
3xy and 5xy are like terms since both terms contain the same variables with the same exponents. 1.3@#$ and 9$#@ are like terms since they contain the same variables (@, #, and $) with the same exponents. Note that the order does not matter. 3 and 3/2 are like terms since there is no variable. (In fact, they are just numbers or constants.)
Nonexamples of Like Terms
3x + y and 4xy are not like terms since 3x + y is actually two terms and x + y is not the same as xy. and 17x are not like terms since the variable x does not have the same exponent in each term. and are not like terms since the terms do not contain the same variable even though the exponents are the same.
Check for Understanding
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External Links to Extend Basic Understanding
Lesson on Writing Algebraic Expressions from Mrs. Glosser: Read more about parts of expressions and practice writing algebraic expressions.
Related Investigations on InterMath to Challenge Teachers
X-Intercept Transformation: Identify the change in the x-intercepts of a function when you reverse its leading coefficient and constant term. Comparing Lines: Examine the relationship between the coefficients in two linear functions if they are parallel, perpendicular, share the same intercept, and more. Coefficients that Affect the Graph: Observe changes in functions as you modify their coefficients. Zero Coefficients: In the equation Ax + By = C, what happens when A, B, or C are zero?
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Evaluating Expressions
Evaluate an Algebraic Expression - To find the value of an expression by replacing each variable in an expression with numbers. Evaluate a Numerical Expression - To perform operations to obtain a single number or value.
Distributive Property - A property of real numbers and many algebraic expressions, where a(b + c) = ab + ac. Often referred to as the "distribution of multiplication across addition." FOIL - A method based on the Distributive Property that is often used to multiply two binomials.
The Distributive Property can be used to multiply two binomials. For example, (x + y)(b + c) means (x + y)b + (x + y)c because of the Distributive Property (look at the definition above and replace a with (x + y).) Then the Distributive Property can be used two more times to get xb + yb + xc + yc. FOIL, described in detail below, is just a way to remember the use of the Distributive Property in this situation. After you read about FOIL, try the procedure on (x + y)(b + c). Do you get the same result as given in this paragraph?
F means you multiply the first two terms together. O means you multiply the outer terms together. I means you multiply the inner terms together. L means you multiply the last two terms together.
In (a + b)(c + d), a and c are the first terms since they appear first in each binomial. a and d are the outer terms since they are on the left and right sides, respectively. b and c are the inner terms since they appear in between the other two terms. b and d are the last terms since they appear second in each binomial.
Examples of FOIL
(3 + x)(2 - 3x) = 3*2 + 3*(- 3x) + x*2 + x*(- 3x) = 6 - 9x + 2x -. Combining like terms, we obtain: 6 - 7x - .
(x - 2)(1 - 5y) = x*1 + x*(- 5y) + (- 2)*1 + (- 2)*(- 5y) = x - 5xy - 2 + 10y. There are no like terms to combine.
(3x + 5)(y - 8)
(a)
(b)
(c)
(d)
xy
Evaluating Expressions at Math.com: Practice evaluating different expressions for given values of the variables.
Multiplying Binomials from regentsprep.org: Read about FOIL and three other methods for multiplying binomials.
From Ask Dr. Math at the Math Forum: Read another explanation of FOIL.
From Ask Dr. Math at the Math Forum: Read another explanation of the Distributive Property.
From Ask Lois at loisterms.com: Read about the Distributive Property.
Dividing Rectangles: Find a possible area of a rectangle that is formed from a larger integral rectangle.
Types of Polynomials
Examples of Monomials
5, 3x, t,
Examples of Binomials
3 + 2b, r - y, 1 + 4p
Examples of Trinomials
3 + t + r, a - b + c, 5 - u - 3v
Nonexamples of Trinomials
3 + 4 + x is not a trinomial since 3 + 4 can be combined into 7. So 3 + 4 + x = 7 + x which is a binomial.
From Ask Dr. Math at the Math Forum: Read one student's question about monomials and polynomials and Dr. Math's response.
X-Intercept Transformation: Identify the change in the x-intercepts of a function when you reverse its leading coefficient and constant term. Coefficients that Affect the Graph: Observe changes in functions as you modify their coefficients. Fundamental Theorem of Algebra: What is the largest number of times a function crosses the x-axis? How about the smallest number? Symmetry of Polynomial Functions: Describe the symmetry of a function.
Types of Expressions
Example of Algebraic Expressions In an algebraic expression, the variable represents values of a quantity that can vary. We need to define what the variable represents. Suppose we wanted to write an expression to find the total cost of movie tickets for different groups of people, given that tickets cost $5 each. We can define "P" to represent the number of people in a group. Then
5 x P or 5P
represents the total cost of movie tickets for a group with P people. So, if our group had 3 people, our algebraic expresion would become
5 x 3 or 15.
Another Example of Algebraic Expressions
Suppose we wanted to write an expression to find the perimeter of the figure below:
We know the length is equal to 75, but the width is not known. We can define "w" to be the width, so that our expression for the perimeter is
w + w + 75 + 75 = 2w + 150
Non-Example of Algebraic Expressions The expression 550 + 20 is not an algebraic expression because it does not contain at least one variable.
We also distinguish between an algebraic expression and an algebraic equation. An equation consists of algebraic expressions and an equal sign. An example of an algebriac equation is shown below.
10x + 3 = 8x - 7
Did you know? Some historians claim that our word "algebra" comes from a book written by al-Khowarizmi, an Arabian mathematician. The title of the book was al-jabr w'al-muqabalah, which means opposition (al-jabr) and restoration (al-muqabalah). So, we may say that al-jabr means to "add the opposite" and al-muqabalah means to "combine your like terms".
Bonesetter and Bloodletter? The Moors took the word al-jabr into Spain, where it took on a nonmathematical meaning. The word algebrista was used to mean a "restorer of broken bones" or one who resets broken bones. Barbers at that time also doubled as "doctors". So, it was not uncommon to see a sign over the entrance to a barber shop that read "Algebrista y Sangrador" (bonesetter and bloodletter).
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