Session 2: Algebraic Expressions

Great minds have long sought to explain the relationship between algebra and music. The closest connection between music and algebra is that they both use patterns. Music is full of patterns. Take a look at this series of numbers: 

                                                                           0,  1,  1,  2,  3,  5,  8,  13,  21,  34, ….

It’s called the Fibonacci sequence, and it’s made up of numbers that are each the sum of the previous two numbers. It’s a sequence that musicians and mathematicians have studied for hundreds of years. There is a link between this sequence and music. 

For example, a piano keyboard takes 13 keys from one C note to the next, eight white and five black. And the black keys are in groups of 3 and 2.  So 13, 8, 5, 3, 2—yes, that’s part of the Fibonacci sequence. 

Converting music to a digital form also requires representing the music mathematically. Algebra is used to capture, analyse and even create music.

So next time when you’re listening to music, try to predict what is next based on what you have just heard in that piece or what you have heard before in pieces of the same style, and so on.

In this unit, we will use algebra tools, such as writing expressions and solving equations to explore patterns in the real world. Recognising a pattern and writing an expression that uses operations and numbers instead of words are two essential skills in algebra. 

In algebra, we generally come across two types of symbols, namely constants and variables.

What is a Constant?

A symbol that has a fixed numerical value. Example: 9, -42, 0, \(\frac{3}{8}\), \(7\frac{1}{3}\) are all constants. 

What is a Variable?

A symbol that can be assigned different numerical values or whose values can vary. It is interesting to note that the variables are known as literal coefficients. Literal means “letter”. Variables are generally represented by a letter of the English alphabet—for example, a, b, c or x, y, etc. However, we avoid using e, i and o because e and i stand for special mathematical terms and o to prevent confusion with zero.

Note: A combination of a constant and a variable, such as 2x or 2+x or 2−x or \(\frac{2}{x}\) are also variables as the value of these terms depends upon the value of x, which can change.

What are Algebraic Expressions?

A combination of constants (−4,0,12…) and variables or literals (a,t,w,…), connected by the signs of fundamental mathematical operations, is called an algebraic expression. For example, 3p−4, 17y, etc., are all algebraic expressions.

Algebraic expressions can be formed by combining variables, other variables, or constants. The different combinations are:

• Combining variables with themselves: The algebraic expression b² is formed by multiplying the variable b with itself. 
• Combining variables with other variables: The algebraic expression (a+b) is formed by adding two variables. At the same time, the algebraic expression ab is formed by just multiplying the two variables. 
• Combining constant and variables: The algebraic expression 6ab is formed by multiplying a constant 6 with the variables a and b.  

  Algebraic Expression:

An algebraic expression is a combination of constant and literals (variables) connected by the signs of fundamental operations. 

Terms of an Algebraic Expression

Various parts of an algebraic expression that are separated by the signs of plus(+) or minus(-) are called the terms of the algebraic expression. For example, in the algebraic expression  and 8 are the two terms of the algebraic expression. Similarly, the algebraic expressions  and −7 are the three terms. A term of an algebraic expression can be a constant, a variable or a product of a constant and a variable.

No matter how complicated an algebraic expression gets, we can always separate it into one or more terms. Terms are really useful because we can follow the rules to combine them and perform the four basic operations.

 Terms of an Algebraic Expression:

The terms of an algebraic expression are parts of the expression, separated by + or – signs.

Factors of a Term

We are familiar with the term factor. The numbers and variables that multiply together to make a term are called factors. We can factorise a term by dividing it into factors (15=5×3). As in numbers, we can also factorise the terms of an algebraic expression. For example, let us find the factors of 2a²−7ab using a tree diagram.

What are Coefficients?

A term generally consists of one numerical factor (constant) the rest variable(s). The numerical factor of a term is the numerical coefficient or simply the coefficient of the term. For example, in the expression −15x²y,−15 is the numerical coefficient, and x²y is the literal coefficient.

A coefficient is a relative term. In a term, any one of the factors is called the coefficient of the product of the other factors. For example, in , the coefficient of x² is 5yz, the coefficient of y is 5x²z, and the coefficient of z is 5x²y.

1) Write the coefficient of a in each of the following algebraic expressions. 

i. Coefficient of a in .

ii. Coefficient of a in .

iii. Coefficient of a in .

Like Terms and Unlike Terms

The terms of an algebraic expression that have the same literal factors- that is, both the letters and their exponents (powers) have to be identical are called like terms. In like terms, the numerical coefficient may be different.

For example, in the algebraic expression  and 2ab are like terms. On the other hand, 5b and −7b² are not like terms even though they both have the same variable b, as here the powers of b are different. 

The terms of an algebraic expression that do not have the same literal factors are called unlike terms. For example, and 7ab² are unlike terms, as the literal factors of the two terms are different. 

 Like and Unlike Terms: 

Terms in the expression with the same literal factors are called like terms. Terms that do not have the same literal factors are called unlike terms. 

Type of Algebraic Expressions

• Monomial is an algebraic expression of one term.
    
  
• Binomial is an algebraic expression of two terms.      

• Trinomial is an algebraic expression of three terms.

An algebraic expression with more than three terms is named by its number of terms. For example, an algebraic expression with five terms is called a five-term algebraic expression. 

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What are Algebraic Expressions? – Examples

Example 1 

Write down the numerical and literal coefficients of each of the following monomials.     

i. Given monomial is −15ab²

    Its numerical coefficient =−15

    Its literal coefficient =ab²

ii. Given monomial is 

    Its numerical coefficient =\(-\frac{7}{2}\)

    Its literal coefficient =

iii. Given monomial is \(-\frac{xy}{2z}\)

    Its numerical coefficient =\(-\frac{1}{2}\)  

    Its literal coefficient =\(\frac{xy}{z}\)

Example 2

Write down the terms of the expression . Also, write down the coefficient of a² in the term .

The given expression has five terms, namely  and 2ac².

The coefficient of a² in the term .

Example 3

Write down the coefficients of x, xy and xyz in the term  of the algebraic expression .

The given algebraic expression has four terms, namely,  and .

Coefficient of x in .

Coefficient of xy in .

Coefficient of xyz in .

Example 4

In the following, write down the pairs which contain like terms 

i. Terms \({{a}^{2}}b\), −14\({{a}^{2}}b\) have the same literal factor \({{a}^{2}}b\). So, pair \({{a}^{2}}b\) and −14\({{a}^{2}}b\) are like terms. 

ii. 5xy and 3z are unlike terms as their literal factors 5xy and 3y are distinct. 

iii. \({{m}^{2}}n\) and 5l\({{m}^{2}}n\) are unlike terms as their literal factors \({{m}^{2}}n\) and 5l\({{m}^{2}}n\) are distinct. 

 Remember this! 

• An algebraic expression is a combination of constant and literals (variables) connected by the signs of fundamental operations. 
• A variable is a symbol that can be assigned different numerical values or whose values can vary. 
• The terms of an algebraic expression are parts of the expression, separated by + or – signs. 
• In a term with numbers and variables, each variable or number is called a factor of that term, and each factor is the coefficient of the product of the remaining factors. 
• Terms with the same literal factors are called like terms, while terms with different literal factors are called unlike terms. 
• A constant is a symbol that has a fixed numerical value.