Algebraic Expressions
A
mathematical statement which is obtained by using mathematical fundamental
operation between constants and variables with different powers is called an
algebraic expression.
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For
example:
- The sum of 2 and x is 2 + x, where the constant
2 and variable x is connected by ‘+’ sign.
- So, 2 + x is an algebraic expression.
- Similarly, other algebraic expressions are
2xy, 3x + 4y, x – y^{2} + 4 , etc.
Algebraic Terms
The parts of an algebraic expression separated by ‘+’
or ‘–’ signs are called algebraic terms.
For
example:
- 3x + 2y is an algebraic expression, where
3x and 2y are two parts connected by ‘+’ sign. So, 3x and 2y are called the
terms of the expression 2x + 3y.
Types of algebraic expression
Algebraic expressions are of different types. They are:
Monomial, Binomial, Trinomial and multinomial expressions.
- Monomial
Expression: An
algebraic expression which contains only one term is called a monomial
expression. For example: 4x, 3x^{2}y, -7xyz^{2}, etc.
- Binomial
Expression: An algebraic
expression which contains two terms connected by ‘+’ or ‘–’ is called a
binomial expression. For example: a + b, 2a^{2} – 3abx^{2}, 3x
– 4ay^{2}, etc.
- Trinomial
Expression: An
algebraic expression which contains three terms connected by ‘+’ or ‘–’ is
called a trinomial expression. For example: x + 2y – z, a^{2} + 2ab + b^{2},
etc.
- Multinomial
Expression: An
algebraic expression which contains four or more than four terms connected by
‘+’ or ‘–’ is called a multinomial expression. For example: 2x – 2y + z, x^{3}
– a^{2} + 2ab – b^{2} + 7, etc.
Coefficient, base and power
- Coefficient: A numerical or
constant quantity placed before the variable and multiplying it is called the
coefficient. For example: In an expression 3x, 3 is called the coefficient of
x.
- Base: An alphabet letter
used in algebraic expression is called the base. For example: In an expressioin
3x^{2}, x is the base.
- Power: The repeatation of
the same variable for the required number of times in an algebraic expression
is called power. For example: In an expressioin 3x^{2}, 2 is the power
of x.
Like and Unlike Terms
- Like
Terms:
The terms are like if the terms have the same base with same power of the
variable. For example: 2a, -3a, 4a are like terms. x^{3}, 2x^{3},
-5x^{3}, -3x^{3} are like terms. similarly 7xy, -3xy, 2xy are
like terms.
- Unlike
Terms:
The terms are unlike if the terms have the different variables or different
powers of the base. For example: x, 2x^{2}, 3x^{2}y, -y are
unlike terms. x^{3}, x^{4}, x^{-3}, -x are unlike
terms. Similarly xy, 2xz, 3x^{2}y are unlike terms.
Values of Algebraic Expressions
When we substitute the value of a variable in an
algebraic expression with a number, the value of the expression is obtained
after use of mathematical fundamental operations, it is called the value of the
expression.
For
example: if
x = 2 and y = 3, find the value of 2x – 3x^{2}y.
Here, the given expression = 2x – 3x^{2}y
= 2 × 2 – 3 × 2^{2} × 3
= 4 – 3 × 4 × 3
= 4 – 36
Workout Examples
Example 1: Write the type of each
expression.
a.
4x
b.
2a
+ 3b
c.
3x^{2}
– 4xy + 2y^{2}
d.
9x^{3}
– 10x^{2} + 4x – 21
e.
3x^{3}
– 2x^{2} + 5x + 7xy - 8
Solution: Here,
a. 4x -------> Monomial
b. 2a + 3b ---------> Binomial
c.
3x^{2}
– 4xy + 2y^{2} ---------> Trinomial
d.
9x^{3}
– 10x^{2} + 4x – 21 ----------> Multinomial
e.
3x^{3}
– 2x^{2} + 5x + 7xy – 8 ----------> Multinomial
Example 2: Rewrite the following
statements in algebraic expressions.
a.
The
sum of twice x and 5.
b.
Three
times the difference of x and y is less than 3a.
c.
5x
is added to the product of b and 7.
d.
One
fourth of x is added to 25
e.
7x
is divided by 14 and added to product of c and 4.
Solution: Here,
a. 2x + 5
b. 3(x – y) < 3a
c.
5x + 7b
d.
x/4 +25
e.
7x/14 +4c
Example 3: Write the terms having:
a.
Base
= x, power = 2 and coefficient = 1
b.
Base
= x, power = 3 and coefficient = -3
c.
Base
= y, power = 5 and coefficient = 4b
d.
Base
= z, power = 2 and coefficient = -a
Solution: Here,
a. x^{2}
b. -3x^{3}
c.
4by^{5}
d.
–az^{2}
Example 4: Identify the like and unlike
terms of the following expressions:
a.
3x,
-2x, 7x
b.
3a^{2},
3a^{3}, 4a
c.
2(x+y),
3(x+y)^{3}, 9(x+y)^{2}
Solution: Here,
a. Like terms
b. Unlike terms
c.
Unlike
terms
Example 5: If x = 4 and y = 3, find the
value of 2xy + 3y^{2}
Solution: Here,
x = 4
y = 3
∴ 2xy + 3y^{2} = 2 × 4 ×3 + 3 × 3^{2}
= 24 + 3 × 9
= 24 + 27
= 51You can comment your questions or problems regarding the algebraic expressions here.
What is the base in the algebraic expression (3x)2
ReplyDeleteBase in algebraic expressions is the term of reference. We may consider 3x as the base while simplifying the expression like (3x)^2 × (3x)^3 × (3x)^4 = (3x)^2+3+4 = (3x)^9 = 19683x^9. Or, we may take x as the base after the second step like (3x)^2 × (3x)^3 × (3x)^4 = 9x^2 × 27x^3 × 81x^4 = 9×27×81x^2+3+4 = 19683x^9.
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