Algebraic Expressions

Algebraic expressions are a super useful part of algebra. It is where we start using letters and symbols to solve problems. It is like a middle ground between basic math with numbers and the tricky stuff in advanced math. In this chapter, we are going to learn the basic ideas of algebraic expressions and how they help us solve all sorts of problems in our everyday lives.

Terms related to Algebraic Concepts

Certainly, there is a simpler explanation of those algebraic concepts:

Algebra

Algebra is a part of mathematics where we use symbols to represent numbers and statements. These symbols, known as literals, can stand for various values and they act like variables.

Variable: A variable is like a symbol that can stand for different numbers. We usually use letters like a, b, c, p, q, r, x, y, z, etc.
Constant: A constant is a symbol that always represents a specific number. We usually use numbers like 3, 1/7, 3?, 0.5, −2, −1½, etc.
Term: When you multiply or divide variables or constants together, you create a term. Examples of terms are 5a, 7abc, 3t, −2xy, −3a/5, a³, −21x²??/11, etc.
Coefficient: In a term, the coefficient is like the number that you get when you multiply all the parts together including the sign.
Algebraic Expression: When you add or subtract two or more terms, you make an algebraic expression.

An algebraic expression 5y² − 4y + 17 has one constant, two variables and three terms.

Like and Unlike Terms: In an algebraic expression, terms with the same letters are like terms. Terms with different letters or different powers of letters are unlike terms. For example, 5xy and −3xy are like terms but 5xy and −3ax are unlike terms.

Example: A taxi charges $27 per km and a fixed charge of $45. If the taxi is hired for z km, which of the following is an algebraic expression to find the total fare?

a) 27z − 45
b) 27z + 45
c) 45z + 27
d) 45z − 27

Answer: b)  27z + 45

Explanation: Charges of a taxi per km = $27
Fixed charge = $45
Taxi is hired for z km.
Charges for z km = $27 × z = $27z
Algebraic expression to find the total fare = Charges for z km + Fixed charge
                                                             = $(27z + 45)

Various Types of Algebraic Expression

Monomial: A monomial is like an algebraic expression that has just one term. Example of monomials are 4x, −7ab, 3b⁵, −2p/5, 0.5q,,−11, etc.
Binomial: A binomial is an algebraic expression that has two terms. Examples of binomials are (2a + 3b), (5x − 7z), (13x² − 8xy³), etc.
Trinomial: A trinomial is an algebraic expression that has three terms. Examples of trinomials are (3a + 2b + 5c), (4p + q − c/2), (x² − 2y³ − z³), etc.
Quadrinomial: A quadrinomial is an algebraic expression that has four terms. Examples of quadrinomials are 11 − x² + y² − 2z, a − 3b + c² + cd, etc.
Polynomial: A polynomial is a type of algebraic expression that includes variables, constants, coefficients, exponents and mathematical operations. The terms within polynomials are the different parts of the expression typically separated by either "+" or "−" signs. All polynomials are a kind of algebraic expression.

Example: Simplify the following.

7xy² − y² + 7x²y − 5x² − 3y² + 4y²x  − 3y² + x²

a) 7xy² + 11xy² − 4x² − 7y²
b) 7xy² − 11xy²  + 4x² − 7y²
c) 11xy² + 7xy² − 4x² − 7y²
d) 11xy² − 7xy²  + 4x² − 7y²

Answer: c) 11xy² + 7xy2 − 4x² − 7y² 

Explanation: 7xy² − y² + 7x²y − 5x² − 3y² + 4y²x  − 3y² + x²
= 7xy² + 4y²x + 7x²y − 5x² + x² − 3y² − y² − 3y²
= 7xy² + 4xy² + 7x²y − 5x² + x² − 3y² − y² − 3y² [Arranging the like terms]
= (7 + 4)xy² + 7x²y − (5 − 1)x² − (3 + 1 + 3)y²
= 11xy² + 7xy² − 4x² − 7y²