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Algebra is one of the most important part of Mathematics.Algebra is used to find out the unknown value of variables or quantity.This can be said that algebraic expression is a process of multiplying two given terms consist of variables and constants.These expression undergoes mathematical operations like multiplication, addition, subtraction and division.Algebraic expression is a type of expression which is formed by the combination of variables and constants. Let's understand this with an example. Ex - (4xy × 8) In this given expression x and y are variables whereas 4 and 8 are constants. Simply, this can be said that the value of algebraic expression is directly dependent on the value of variables.The whole value of expression changes according to the given constants.
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What is an Algebraic Expression
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Algebraic expressions is a set of expressions which consist of variables ,constants,integers and exponents.The expression undergoes various mathematical operations like addition,subtraction,multiplication and division.This process is called algebraic expressions. The algebraic expressions contain one or more than one terms.This is known as polynomial expression.
To understand an algebraic expression,the example is given.
Ex- 2x+1 If the value for x = 1, then the value of the given expression will be 3. If there is a value of x = 2 then the value will be 5.So it can be said that the value of the algebraic expression is totally dependent on the given value of variables.
Terms in Algebra
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The algebraic expression consists of many different terms together.So we will discuss those terms one by one. Let's get started with it.
Variables are those terms that do not have any mathematical value.We have to find out with the help of mathematical operations.They are unknown quantities and they are written as x, y, z, a, b, c, f etc.
Coefficient is a value which is united with the variable in the expression.For ex- 4zy ,in this expression the coefficient is 4 i. e, numerical value.
Constants are defined as the numerical value in an algebraic expressions.For ex- 7z + 4 , in the above expression 4 is the numerical value and said to be constant.
Polynomial Expression -The algebraic expression consist of one or more terms and are made from the combination of variables, coefficients, exponents, and constants is called a polynomial expression.On the basis of the term, there are three types of polynomial expression.They are discussed below.
- Monomial Expression:Those expressions which have only one term are known as monomial expression.Example - 3x, 5y,7z etc.
- Binomial Expression: Those expressions which have two terms are known as binomial expression.Example - 3x+6, 4(2+y), 6x-8.
- Trinomial Expression: Those expressions which have three terms together are known as trinomial expression.Example - 5xy+2x+8, 9x + y - 5.Here the expression contains three terms together.
Multiplication of Algebraic Expression
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The multiplication of algebraic expressions are done in the same way integers are multiplied along with the concepts of how like and unlike terms are multiplied. Accordingly we must know the basic properties of arithmetic operations.
The multiplication of different types of algebraic expressions discussed below.
As we are now aware of the key terms and what algebraic expressions really means.Now let's toddle upon another concept named multiplication of algebraic expressions.
The multiplication of these expressions seems to be complicated for the young folks but a proper understanding will really help them out. Let's have a glimpse over this.
The methods used to multiply the integers and perform other operations on arithmetic is the core of solving the aforementioned problems. The foundation understanding of the mathematical concepts like different arithmetic calculations, properties of different types of numbers,use of like and unlike terms concepts are used. The different types of algebraic expressions follow different rules of multiplication.
Understanding in a bit more detail
- Multiplying monomial with a monomial:
The single term expressions are known as monomials . For multiplying monomials we have to arrange the like terms and multiply accordingly. An example would help you out .
Ex.1. 3x × 4y
= 3 × x × 4 × y
= 3 × 4 × x × y
= 12 × x × y
= 12xy
Here in this example we can clearly see that the like terms are arranged first and then multiplied accordingly.This eases our calculation and helps proper understanding.
Some more examples are given below.
Ex.2 5x² × 6x = 30 x³
Ex.3 -12ab³ × 15ac² = -180a²b³c²
Multiplying monomial with a polynomials
Here in this segment we would understand how to multiply a monomial and a polynomial. This calculation requires the concept of distributive property of multiplication over addition or subtraction.
It would be easy to understand using some examples.
Ex.1 5a² × (6x + 9y)
= 5a² × 6x + 5a² × 9y
= 30a²x + 45a²y
Ex.2 (9b + 61a) ×2ab
= 18ab² + 122a²b
As done above the distributive property over addition is used in this example.Usually we use the concepts of distributive property over addition or subtraction .
Multiplying polynomials with a polynomial
Getting started with the topic, the polynomials are those algebraic expressions with 3 or more than 3 terms in it.And to multiply these polynomials we have to do a bit more calculation for solving it.Though it also works on the concepts of distributive property.
What we should do while solving multiplication of two polynomials are as discussed below.
Ex.1 ( 5m +6q) × (7n -8d)
= 5m × 7n - 5m × 8d + 6q ×7n - 6q ×8d
= 35mn - 40md + 42nq - 48dq
Well that's the required answer as we don't have any more like terms to do calculations afterwards. Here we see that firstly we used the distributive property and then multiplied accordingly. Afterwards check for any like terms of any to proceed further in our calculations.
Thus while we are multiplying polynomials with polynomials we have to keep these concepts in mind.
Things to Remember
- Algebraic expression is a set of expressions that consists of variables, constants,base and exponents with it.For ex- 3x, 4zy+6, 5z+3 etc.
- Algebraic expression contains one or more than one terms in it.The involvement of one or two terms in an expression is known as polynomial expression.
- Variables are the unknown quantity in algebra.Variables are used to find out the unknown value of the given symbol.The symbols are written as x, y, z, a, b, c etc.
- Constants is a term which can be defined as the value or number that cannot be changed in the given expression.It is known as constants.Ex-
4c + 5 ,In this expression 4 and 5 are
numbers that show numerical value.
- Multiplication of algebraic expression are the set of expression which is simplified by the mathematical operation i.e.multiplication.For ex- 8zy
In this expression, there is a hidden sign of multiplication in between 8 and z, z and y.So when it is calculated then the product will be 8zy.
- The value of an algebraic expression changes according to the assigned value of variables.Suppose there is an expression given 4xz where the value of x=2 and z=1 are given.Then,
(4)×(x)×(z)
4×2×1 = 8 will be the answer.The whole expression changed when we put the value of x and z.
Sample Questions
Ques. What is an algebraic expression? (2 Marks)
Ans. Algebraic expression can be defined as the set of expressions consisting of different terms together such as variables, constants, base,exponent etc.This is known as algebraic expression. Ex- 10x, 2y+z etc.Here it can be seen that the expression has different components in it.
Ques.Solve the algebraic expression. (2x+3y) (4a+6b). (2 Marks)
Ans. In this expression, we have to solve it by using the distributive property of multiplication.
A/C to distributive law,
(2x+3y) (4a+6b)= 2x(4a+6b) + 3y(4a+6b)
=8xa + 12xb + 12ya + 18 yb
The whole expression was solved according to this formula, a(b + c) = ab + ac.
Ques.What are variables in the algebra? (2 Marks)
Ans. Variables can be defined as the alphabets or symbols that are used in an algebraic expression that shows a specific value of it. This is known as variables.The example to understand this is given here.
6zy + 4 , In this algebraic expression 'z' and 'y' are two alphabets having different values.
Ques.What do you mean by constants in the algebra? (2 Marks)
Ans. Constants are the numerical value in the algebraic expression.This is termed as constants.To understand this an example is given- 4x - 6y + 6,in the given algebraic expression, the constant is 6.
Ques.What are the different terms used in algebra? (2 Marks)
Ans. The algebraic expressions are made from the combination of various terms.The terms such as constants, variables, base, exponent and coefficient.Each term has some specific value.
Ques.Solve the algebraic expression given. (3z+5) (5x+6) and (5a*a). (2 Marks)
Ans. The first expression given is,
(3z+5) (5x+6) = 3z (5x+6)+ 5(5x+6)
[A/C to distributive law of multiplication]
= 15 zx + 18z + 25x + 30 Ans.
(5a*a) , In this expression we have to multiply the variable i.e. 'a' with 'a'
(5a×a) = 5a² Ans.
Ques.What is a polynomial expression? (1 Marks)
Ans. Polynomial expressions are the part of algebraic expression that is made by the combination of variables and constants.The polynomial expressions contain one or more than one terms in it and undergo mathematical operations.
Ques.Mention the types of polynomial expression. (3 Marks)
Ans. There are three types of polynomial expressions based on the terms.They are discussed below.
- Monomial expressions are the type of polynomial expression which have only one term in it.This is called as monomial expression.Ex- 5z, 4ac
- Binomial Expression is a type of polynomial expression which contains two terms together.This is known as binomial expression.For ex- 2z+3, 7x+6 etc.
- Trinomial Expression is a type of polynomial expression which contains three terms together.This is known as
trinomial expression.Ex-6x + 3y - 4,5ab + 4c + 8 etc.
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