Mathematics (Part 1:All About Algebraic Expression)

Algebraic expression

"Rational expression" redirects here. For the notion in formal languages, see regular expression.

In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number).^[1] For example, ${\displaystyle 3x^{2}-2xy+c}$ is an algebraic expression. Since taking the square root is the same as raising to the power ${\displaystyle {\tfrac {1}{2}}}$,

${\displaystyle {\sqrt {\frac {1-x^{2}}{1+x^{2}}}}}$

is also an algebraic expression. By contrast, transcendental numbers like π and e are not algebraic.
A rational expression is an expression that may be rewritten to a rational fraction by using the properties of the arithmetic operations (commutative properties and associative properties of addition and multiplication, distributive property and rules for the operations on the fractions). In other words, a rational expression is an expression which may be constructed from the variables and the constants by using only the four operations of arithmetic. Thus, ${\displaystyle {\frac {3x^{2}-2xy+c}{y^{3}-1}}}$ is a rational expression, whereas ${\displaystyle {\sqrt {\frac {1-x^{2}}{1+x^{2}}}}}$ is not.
A rational equation is an equation in which two rational fractions (or rational expressions) of the form ${\displaystyle {\frac {P(x)}{Q(x)}}}$ are set equal to each other. These expressions obey the same rules as fractions. The equations can be solved by cross-multiplying. Division by zero is undefined, so that a solution causing formal division by zero is rejected.

1. Addition and Subtraction of Algebraic Expressions

Before we see how to add and subtract integers, we define terms and factors.

Terms and Factors

A term in an algebraic expression is an expression involving letters and/or numbers (called factors), multiplied together.

Example 1

The algebraic expression

5x

is an example of one single term. It has factors 5 and x.
The 5 is called the coefficient of the term and the x is a variable.

Example 2

5x + 3y has two terms.

First term: 5x, has factors $\displaystyle{5}$5 and x
Second term: 3y, has factors $\displaystyle{3}$3 and y

The $\displaystyle{5}$5 and $\displaystyle{3}$3 are called the coefficients of the terms.

Example 3

The expression

$\displaystyle{3}{x}^{2}-{7}{a}{b}+{2}{e}\sqrt{{\pi}}$3x​2​​−7ab+2e√​π​​​

has three terms.

First term: $\displaystyle{3}{x}^{2}$3x​2​​ has factors $\displaystyle{3}$3 and x²
Second term: $\displaystyle-{7}{a}{b}$−7ab has factors $\displaystyle-{7}$−7, a and b
Third Term: $\displaystyle{2}{e}\sqrt{{\pi}}$2e√​π​​​; has factors $\displaystyle{2}$2, $\displaystyle{e}$e, and $\displaystyle\sqrt{{\pi}}$√​π​​​.

The $\displaystyle{3}$3, $\displaystyle-{7}$−7 and $\displaystyle{2}$2 are called coefficients of the terms.

Like Terms

"Like terms" are terms that contain the same variables raised to the same power.

Example 4

3x² and 7x² are like terms.

Example 5

-8x² and 5y² are not like terms, because the variable is not the same.

Adding and Subtracting Terms

Important: We can only add or subtract like terms.
Why? Think of it like this. On a table we have 4 pencils and 2 books. We cannot add the 4 pencils to the 2 books - they are not the same kind of object.
We go get another 3 pencils and 6 books. Altogether we now have 7 pencils and 8 books. We can't combine these quantities, since they are different types of objects.
Next, our sister comes in and grabs 5 pencils. We are left with 2 pencils and we still have the 8 books.
Similarly with algebra, we can only add (or subtract) similar "objects", or those with the same letter raised to the same power.

Example 6

Simplify 13x + 7y − 2x + 6a 
~answer:
13x + 7y − 2x + 6a
The only like terms in this expression are $\displaystyle{13}{x}$13x and $\displaystyle-{2}{x}$−2x. We cannot do anything with the $\displaystyle{7}{y}$7y or $\displaystyle{6}{a}$6a.
So we group together the terms we can subtract, and just leave the rest:

(13x − 2x) + 6a + 7y
= 6a + 11x + 7y

Types of Algebraic Expressions

Types of algebraic expressions may further be distinguished in the following five categories.

They are: monomial, polynomial, binomial, trinomial, multinomial.

1. Monomial: An algebraic expression which consists of one non-zero term only is called a monomial.
Examples of monomials:
a is a monomial in one variable a.
10ab² is a monomial in two variables a and b.

5m²n is a monomial in two variables m and n.

-7pq is a monomial in two variables p and q.

5b³c is a monomial in two variables b and c.

2b is a monomial in one variable b.

2ax/3y is a monomial in three variables a, x and y.

k² is a monomial in one variable k.
2. Polynomial: An algebraic expression which consists of one, two or more terms is called a polynomial.
Examples of polynomials:
2a + 5b is a polynomial of two terms in two variables a and b.
3xy + 5x + 1 is a polynomial of three terms in two variables x and y.
3y⁴ + 2y³ + 7y² - 9y + 3/5 is a polynomial of five terms in two variables x and y.

m + 5mn – 7m²n + nm² + 9 is a polynomial of four terms in two variables m and n.

3 + 7x⁵ + 4x² is a polynomial of three terms in one variable x.

3 + 5x² - 4x²y + 5xy² is a polynomial of three terms in two variables x and y.

x + 5yz – 7z + 11 is a polynomial of four terms in three variables x, y and z.

1 + 2p + 3p² + 4p³ + 5p⁴ + 6p⁵ + 7p⁶ is a polynomial of seven terms in one variable p.
3. Binomial: An algebraic expression which consists of two non-zero terms is called a binomial.
Examples of binomials:
m + n is a binomial in two variables m and n.
a² + 2b is a binomial in two variables a and b.

5x³ – 9y² is a binomial in two variables x and y.

-11p – q² is a binomial in two variables p and q.

b³/2 + c/3 is a binomial in two variables b and c.

5m²n² + 1/7 is a binomial in two variables m and n.
4. Trinomial: An algebraic expression of three non-zero terms only is called a trinomial.
Examples of trinomial:
x + y + z is a trinomial in three variables x, y and z.
2a² + 5a + 7 is a trinomial in one variables a.

xy + x + 2y² is a trinomial in two variables x and y.

-7m⁵ + n³ – 3m²n² is a trinomial in two variables m and n.

5abc – 7ab + 9ac is a trinomial in three variables a, b and c.

x²/3 + ay – 6bz is a trinomial in five variables a, b, x, y and z.
5. Multinomial: An algebraic expression of two terms or more than three terms is called a multinomial.
Note: binomial and trinomial are the trinomials.

Examples of multinomial:
p + q is a multinomial of two terms in two variables p and q.
a + b + c is a multinomial of three terms in three variables a, b and c.
a + b + c + d is a multinomial of four terms in four variables a, b, c and d.
x⁴ + 2x³ + 1/x + 1 is a multinomial of four terms in one variable x

a + ab + b² + bc + cd is a multinomial of five terms in four variables a, b, c and d.

5x⁸ + 3x⁷ + 2x⁶ + 5x⁵ - 2x⁴ - x³ + 7x² - x is a multinomial of eight terms in one variable x.

Like and Unlike Terms

In this section we will discuss about like and unlike terms in solving algebraic expressions.

Like terms : The terms having the same literal (variable) with same exponents are called Like terms.

Example: 1) 12x and -5x 2) 4x ² and ½ x ²

Unlike terms : The terms having the same variable with different exponents or different variable with same exponents are called Unlike terms .

Example : 1) 5x and 5y 2) 2x ² and 3y ²

Some more examples :

1)In the algebraic expression 5x ² y + 7xy ² -3xy – 4yx ²

Like terms	5x2y and – 4yx2
Unlike terms	7xy2 and -3xy

2)In the algebraic expression a ² - 3b ² + 7b ² - 9a ² + 6ab + 5

Like terms	Unlike terms
a2 and - 9a2	a2and 6ab
- 3b2 and 7b2	7b2

3)

Like terms	Unlike terms	Reason
	16x; 16y	Variables are not same
x2y ; -7x2y	-	Variables as well as exponents are same
-	9ab ; - 6b	1st term contains ab and 2nd term contains 'b' only

Variable

A symbol for a number we don't know yet. It is usually a letter like x or y.

Example: in x + 2 = 6, x is the variable.

 

Numerical Coefficient

Definition of Numerical Coefficient

The constant multiplicative factors attached to the variables in an expression are known as Numerical Coefficient.

More About Numerical Coefficient

The Numerical Coefficient is always written in front of the variable as shown in the expression given below:
a₁x₁ + a₂x₂ + ................a_nx_n where a₁,a₂...................,a_n are numerical coefficients.
Numerical Coefficient is more frequently referred as Coefficient.

Example of Numerical Coefficient

The numerical coefficient for the term 10x⁴ is 10.
The numerical coefficients for the expression 3x² + x + 1 are 3, 1, and 1.