In mathematics, an algebraic expression is an expression built up from integer constants, variables, and algebraic operations. This article is about Monomials, Binomials, and Polynomials.
An algebraic expression that contains only one non-zero term is known as a monomial. A monomial is a type of polynomial, like, binomial and trinomial, which is an algebraic expression having only a single term, which is a non-zero. It consists of only a single term which makes it easy to do the operation of addition, subtraction, and multiplication.
- g is a monomial in one variable.
- 9cb2 is a monomial in two variables c and b.
- 3a2b is monomial in two variables a and b.
- 4ab/5m is monomial in three variables a, b, m.
- -2m is a monomial in one variable m.
The different parts of a monomial expression are:
- Variable: The letters present in the monomial expression are variables.
- Coefficient: The number before a variable or the number multiplied by a variable in the expression.
- Literal Part: The alphabets which are present along with the exponent values are the literal part.
Let us consider an example 6xy2 is a monomial expression,
- Coefficient is 6
- Variables are x and y
- Degree of monomial expression = 1 + 2 = 3
- The literal part is xy2
The sum of exponent values of variables in the expression is called the degree of monomial or monomial Degree. If variables don’t have any exponent values its implicit value is 1.
4xy3, In this exponent value of x is 1.
degree of expression is 3 + 1 = 4.
The arithmetic operations which are performed on the monomial expression are addition, subtraction, multiplication, and division.
Addition of Two monomials:
When we add two monomials with the same literal part, it will result in a monomial expression.
Addition of 2xy + 4xy = 6xy
Subtraction of Two monomials:
When we subtract two monomials with the same literal part, it will result in monomial expression.
Substraction of 6ab – 4ab = 2ab.
Multiplication of Two monomials:
When we multiply two monomials with the same literal part, it will result in monomial expression.
Product of 2a2b * 6x = 12a2bx
Division of Two monomials:
When we divide two monomials with the same literal part, it will result in a monomial expression.
Division of x6 by x2 = x4
An algebraic expression that contains two non zero terms is known as a binomial. It is expressed in the form axm + bxn where a and b number, x is variable, m and n are nonnegative distinct integers.
- g + 3m is a binomial in two variables g and m.
- 3a4 – 5b2 is a binomial in two variables a and b.
- -4x2 – 9y is a binomial in two variables x and y.
- a2/4 + b/2 is a binomial in two variables a and b.
Any equation that contains one or more binomials is known as a binomial equation.
v = u + 1/2 at2
Operations on Binomials
Few basic operations on binomials are
- Raising to the nth power
- Converting to lower-order binomials
A binomial can be expressed as the product of the other two.
a2 – b2 can be expressed as (a + b) (a – b).
Two binomials can be added if both contain the same variable and the same exponent.
(2a2 + 3b) + (4a2 + 5b) = 6a2 + 8b
It is similar to addition, two binomials should contain the same variable and exponent.
(6a2 + 3b) – (2a2 + 5b) = 4a2 – 2b
When we multiply two binomials distributive property is used and it ends up with four terms. In this method, multiplication is carried by multiplying each term of the first factor to the second factor.
(ax + b) (mx + n) can be expressed as amx2 + (an + mb) x + bn
Raising to nth Power:
A binomial can be raised to the nth power and expressed in the form of (x + y)n
Converting to Lower order binomials:
Higher-order binomials can be factored to lower-order binomials such as cubes can be factored to products of squares and another monomial.
a3 + b3 can be expressed as (a + b) (a2 – ab + b2).
An algebraic expression that contains one, two, or more terms are known as a polynomial.
- 3a + 4b is a polynomial of two terms a and b.
- 2a3 + 3b2 + 4m – 5x + 6k is a polynomial of five terms in five variables .
- a + 2a2 + 3a3 + 4a4 + 5a5 + 6a6 is a polynomial of six terms in one variable.
Types of Polynomials
- Monomial: An algebraic expression that contains only one non-zero term is known as a monomial. A monomial is a type of polynomial, like, binomial and trinomial, which is an algebraic expression having only a single term, which is a non-zero.
- Binomial: An algebraic expression that contains two non zero terms is known as a binomial. It is expressed in the form axm + bxn where a and b number, x is variable, m and n are nonnegative distinct integers.
- Trinomial: An algebraic expression that contains three non-zero terms is known as the Trinomial. For example, a + b + c is a trinomial in three variables a, b and c.
Degree of a Polynomial
In the polynomial equation, the variable having the highest exponent is called a degree of the polynomial.
3a5 + 4a3 – 2a + 6
The degree of above polynomial is 5.
The standard form of representing a polynomial equation is to put the highest degree first and constant term at last.
x4 + 2x3 + 3x2 + x + 5
We can easily solve polynomials using basic algebra and factorization concepts, generally, while solving polynomials’ first step is to set the right-hand side to 0.
Solving Linear Polynomial:
- The first step is to isolate variable term
- Next, make the equation equal to 0
- Solve it using basic algebra operations.
Example: Solve 4a – 8?
4a – 2 = 0
=> 4a = 8
=> a = 8 / 4
=> a = 2
Solving Quadratic Polynomial:
- The first step is to rewrite the expression in descending order of degree.
- Next, equate it to 0
- Perform polynomial factorization.
Example: Solve 4a2 – 4a + a3 – 16?
Rearranging, a3 + 4a2 – 4a – 16
=> a3 + 4a2 – 4a – 16 = 0
=> a2 (a + 4) – 4 (a + 4) = 0
=> (a + 4) (a2 – 4) = 0
Solution is a = -4 and a2 = 4
Multiplication of monomials
Example: Multiply 4a and 3ba3?
First we need to group Coefficients and Variables
(4 * 3) (a * a3) (b)
Apply exponential law,
Multiplication of three or more monomials
Example: Multiply a2, 2ab3, 4ab?
(4 * 2) (a2 * a * a) (b3 * b)
Multiplication of monomial by binomial
Example: Multiply 2a by a + 4?
(2a * a) + (2a * 4)
2a2 + 8a
Multiplication of monomial by trinomial
Example: Multiply 3a by 2a2 + 3ab + 4?
(3a * 2a2) + (3a * 3ab) + (3a * 4)
6a3 + 9a2b + 12a
Multiplication of Binomial by a Binomial
Example: Multiply 4a + 3 and 2a +1?
4a (2a + 1) + 3 (2a + 1) [ now its like multiplication of monomial and binomial ]
8a2 + 4a + 6a + 3
8a2 + 10a + 3
Multiplication of Binomial and Trinomial
Example: Multiply 4a + 1 and a2 + 2a + 1?
4a (a2 + 2a + 1) + 1 (a2 + 2a + 1)
4a2 + 8a + 4a + a2 + 2a + 1
5a2 + 14a + 1
Multiplication of polynomial and monomial
Example: Multiply a3 + a2 + a + b + 3 and 4a?
(4a * a3) + (a2 * 4a) + (a * 4a) + (b * 4a) + (3 * 4a)
4a4 + 4a3 + 4a2 + 4ab + 12a
Multiplication of polynomial and polynomial
Example: Multiply 2x4 + 3x5 + 4 and 2x + 1?
(2x4 * 2x) + (3x5 * 2x) + (4 * 2x) + (2x4 * 1) + (3x5 * 1) + (4 * 1)
2x5 + 6x6 + 8x + 2x4 + 3x5 + 4
6x6 + 5x5 + 2x4 + 8x + 4