Chapter 12: Algebraic Expressions

Chapter -12

Algebraic expressions

Introduction to algebraic expressions

Algebra is the branch of mathematics that deals with the methods of finding the unknown values by using the variables and constants in the algebraic expression.
Algebraic expressions:
Combine variables and constants using the mathematical operations addition and subtraction to construct algebraic expressions.
Terms are either variables, numbers or variables multiplied together with numbers. An algebraic operation like addition and subtraction separates terms.

Variable: A variable is a symbol for an unknown value. It is usually denoted in a letter like x, y, a, b.

The number of terms: The total number of terms is known as the number of terms of a given expression.

Constant: A number which stands alone without any variables is known as constant.

Coefficient: A co-efficient can either be a numerical factor or an algebraic factor or product of both that is a number used to multiply a variable.

E.g. Kathir went to a shop to buy chocolate. He gave to shop keeper ₹50 and the shopkeeper gave him 10 chocolates.Now can you find the cost of one chocolate?

Let x be the cost of one chocolate. Then,

 x ×10=50x = 5010 = 5.

Therefore the cost of one chocolate is ₹5.

Types of algebraic expressions. Monomials

A product of numbers, variables and their exponents is called a monomial.            

A monomial is in a standard form, if first there is a constant (numerical) factor, and then variable factors in alphabetical order. If there are several variable factors with the same base, you multiply them, thus you get one factor for each variable.

A monomial is in a standard form, if:
 •   every product of the same variables is written as one variable with an exponent
am⋅ an=am+n;
• A constant factor (or the monomial coefficient) is written as the first term of a monomial.
The standard form of the monomial (6xy²⋅(−2)x3y) is −12x4y³ 
A constant factor of a standard form of a monomial is called the monomial coefficient.
Monomials that have the like products of the variable terms, even if the terms' order is different, are called similar monomials.
If similar monomials have the like coefficients, then these monomials are called mutually equal monomials.
Similar monomials are the monomials that have the like products of the variable terms, even if the terms' order is different.

To add or subtract monomials, the following steps must be followed:
1) add or subtract monomial coefficients; 
2) do not change the variable factors.

When multiplying or dividing a monomial by a number, the coefficient of a given monomial is multiplied or divided, but the variable factors remain the same.

2) Divide the monomial 14b by 7 
14b:7=(14:7)b=2b
(if 14 bananas are divided equally between 7 people, each person will have 2 bananas)

3) Divide the monomial c by 10  
c:10=(1:10)c=110c=0,1c
(1 lemon divided by 10 is 1 tenth of a lemon)

Types of algebraic expressions. Binomials

Any algebraic expression with two terms is called a binomial expression. a binomial expression has only two terms. This is the reason why it is called a binomial (prefix 'bi' stands for 'two').
Degree in an algebraic expression is the sum of the exponents of the variables in the expression.
Example: 7xy²−3y²z  here

When multiplying or dividing a monomial by a number, multiply or divide only its coefficient.

Example:

6x⋅5=30x;−0,5zy⋅2=−zy;4x²:2=2x²
A binomial is an algebraic sum of monomials.

Example:

3x+2y;4xy−6y²
When multiplying or dividing a binomial by a number, each term is multiplied or divided.

Types of algebraic expressions. Trinomials

A trinomial algebraic expression is an algebraic expression that has three terms.
The prefix 'tri' in 'trinomial' stands for 3 terms.
To add two trinomials, you must:
 1) remove the brackets (without changing the signs, because the “+” sign is in front of the brackets);
2) add the like terms.

Example:
Add trinomials: 

(−5x³+3y−5y²)+(8x³+5y²−2y)

1) Remove the brackets:
(−5x³+3y−5y²)+(8x³+5y²−2y)==−5x³+3y−5y²+8x³+5y²−2y.

2) Find the like terms and add:
−5x³¯¯¯¯¯¯¯¯+3y−5y²+8x³¯¯¯¯¯¯¯¯+5y²−2y=3x³+3y¯¯¯¯¯¯−5y²+5y²−2y¯¯¯¯¯¯=3x³+y.
To multiply a trinomial by a trinomial, you need to multiply each term of one trinomial by each terms of another trinomial and add the resulting products. 
(a+b+c)⋅ (d+e+f)==a⋅d + a⋅e +a⋅f +b⋅d +b⋅e +b⋅f +c⋅d +c⋅e +c⋅f = ad + ae + af + bd + be + bf + cd + ce + cf

To subtract two trinomials , you must:

1) remove the brackets and change the signs of the trinomials that are preceded by the sign "−" to the opposite;

2) combine the like terms of the trinomials.

Example:

Let us calculate the difference of trinomials (7x²+3x−2) and −2x²+2x+3

1) Write down the difference of the trinomials and remove the brackets, taking the signs before the brackets into account:
(7x²+3x−2)−(−2x²+2x+3)=7x²+3x−2+2x²−2x−3

2) Find the like terms:
7x²¯¯¯¯¯+3x¯¯¯¯¯¯¯¯−2 +2x²¯¯¯¯¯−2x¯¯¯¯¯¯¯¯−3

3) Combine the like terms:
7x²¯¯¯¯¯+3x¯¯¯¯¯¯¯¯¯¯¯¯−2+2x²¯¯¯¯¯¯¯¯−2x¯¯¯¯¯¯¯¯¯¯¯¯−3=(7+2)x²+(3−2)x−2−3=9x²+1x−5

4) If the coefficient of a term is 1, then usually we do not write it:
9x²+1x−5=9x²+x−5

Types of algebraic expressions. Polynomials

Polynomial (poly - 'many', nomial - 'term').

 When an algebraic expression has one or many terms, then that expression is called a polynomial expression.
In some cases, polynomial multiplication can be performed easier when using the abridged multiplication formulas.

You need to remember 3 formulas:

1) The square of the sum of two expressions:
(a+b)²=a²+2ab+b²

2) The square of the difference of two expressions:
(a−b)²=a²−2ab+b²

3)  The difference of squares of two expressions:
(a−b)(a+b)=a²−b²

Addition and subtraction of algebraic expressions

The numeric expression refers to any record of numbers, signs of arithmetic operations and brackets, made up with meaning.

For example:
3 + 5⋅(7−4) is an numeric expression.
3+:−5 is not a numerical expression.
An algebraic expression is a record of letters, signs of arithmetic operations, numbers and brackets, made up with meaning.
For example: a²−3b is an algebraic expression.
Whenever it comes to adding algebraic terms, we add the coefficient of like terms together. i.e. coefficient of the variable with its like variable co-efficient and constant with constant.

Addition laws

1) The amount does not change from a change in the places of the terms, i.e.
 a+b=b+a

This is the translational law of addition.
2)To add the third term to the sum of two terms, we can add the sum of the second and third terms to the first term, i.e.
 (a+b)+c=a+(b+c)

1)  This is the combined law of addition.
variable coefficient with its similar variable coefficient and constant with constant except that the fact we will include the term additive inverse.

Example:
To subtract  (12a + 7) from  (30a − 2), we have to add the additive inverse of (12a+7) with (30a−2).
Additive inverse of (12a + 7) is −(12a + 7)=(−12a − 7) 
Therefore,
(30a−2)−(12a + 7)=(30a −2) + (−12a −7)=(30a −12a)+(−2 −7)=18a −9