Variable Definition

Introduction

A variable is used to represent an unknown value in an equation. In algebra, we can use the four basic operations addition (+), subtraction (-), multiplication (×), and division (÷) same as arithmetic. In algebra, terms are mathematical expressions that are made of two different parts: the number part and the variable part. In a term, the number part and variable part are multiplied together and written without a multiplication symbol. Terms can have any number of variables

Expressions

An expression consisting of one or more terms in which variables may have anything as power, including positive, negative, or fractions.

For example, $\mathrm{3x^2-5x +2,7+2x^{-1},3 + x^{\frac{1}{2}}}$, etc.

An expression cannot be expressed with an equal symbol in it. The different parts of algebraic expressions are

Part	Meaning	Example
Coefficient	The numerical part comes before the variable	4x,3a 4 and 3 are coefficient.
Variable	Variable	4x,3a x and a are variables.
Constant	The number with no variable.	4x+5,3a+2 5 and 2 are constant.

Equations

An equation is a mathematical statement where two sides are equal to each other. In algebraic equations, we can find out the unknown values by equating them to the known values. From the given algebraic equations, we need to simplify and rearrange the equation to find the unknown values.

An algebraic equation has, the same variable that cannot be used for different unknowns in the same problem. We can use two different letters to represent the same value.

Polynomials

A polynomial is an algebraic equation containing one or more terms in which all variables have only positive powers.Polynomial is made of any number of terms; it is like a chain of terms connected with an addition or subtraction. The degree of a term is determined by the power of the variable part.

When there are two or more degrees, add up the powers of each variable to get the degree of the term. Polynomials are often referred to by the degree of their highest term. The general form of the polynomial is

$$\mathrm{a_n x^n+a_{n-1} x^{n-1}+a_{n-2} x^{n-2}......a_1 x^1+a_0}$$

Polynomial is classified based on its degree,

Degree	Type	Description	Example
Degree 1	Linear polynomial	1 is the highest power in the polynomial	4x,3a
Degree 2	quadratic polynomial	2 is the highest power in the polynomial	x2+2x.
Degree 3	cubic polynomial	3 is the highest power in the polynomial	x3+x2+x
Degree 0	constant polynomial	0 is the highest power in the polynomial	60 = 6

Zeros of a polynomial p(x) is a number ‘a’ such that p(x) = 0

Variables

A variable is a symbol or alphabet that represents a mathematical quantity that is either arbitrary or unknown. Letters like x, y and z are used to represent variables. A variable represents the unknown values. A variable may only have one value, or it can change depending on the context or situation. For example, Two times something equals ten. Here 2x = 10 where x = 5. We can conclude two times of five equals ten.

In polynomial classified based on number variables such as

• polynomial with one variable Example 5x²-6x+2

• polynomial with two variables Example: 2x + 4y

• polynomial with two variables Example: 2x + 4y + 7z

Solved Examples

1)Identify the terms, coefficients and constants in a given expression 112x+45

Answer: Terms = 112x, 45

Coefficient = 112

Constant = 45

2)Evaluate the expression 1.5x^2-2 when x =5.

Answer: 1.5x²-2 = 1.5(5)²-2

= 1.5 (25) - 2

= 37.5 - 2 = 35.5

3)If 12 + 3x is 9 more than 12. What is the value of 6x?

Answer: 12 + 3x = 9 + 12

12 + 3x = 21

3x = 21 - 12

3x = 9

x = 3

now the value of 6x = 4 (3)

6x = 12

4)Add the given polynomials (4x²+3x+6) and (3x²+5x+12)

Answer : (4x²+3x+6)+(3x²+5x+12) = 4x²+3x+6+3x²+5x+12

= 7x²+8x+18

5)Subtract the given polynomials (11x²+6x+25) and (4x²+3x+13)

Answer: (11x²+6x+25)- (4x²+3x+13) = 11x²+6x+25-4x²-3x-13

= 7x²+3x+12

6)Divide the polynomials x²+7x+12 by x + 3

Answer: $\mathrm{\frac{x^2+7x+12}{x+3} =\frac{(x+3)(x+4)}{x+3}}$

Cancel the common terms we get,

$$\mathrm{\frac{x^2+7x+12}{x+3} = x+4}$$

7)Multiply the polynomials x²+4x+3 and x + 2

Answer: (x²+4x+3 )(x+2) =x³+4x²+3x + 2x²+8x+6

= x³+ 6x²+11x+6

8) Check if x =3 is a zero of the given polynomial x²-3x

Answer: p(x) = x²-3x

when x = 2 we get,

p(2) = (3)²-3(3)

= 9 - 9

= 0

9)What is the remainder if x³+ 6x²+3x-5 is divided by x +2 ?

Answer: Equating the divisor = 0

x +2 = 0

x = -2

Substituting in the given polynomial we have

x³+ 6x²+3x+5 = (-2)³+ 6(-2)²+3(-2)-5

= -8 +24 -6 -5

= 24 - 19

= 5

Conclusion

We have different names to represent different parts of the equation, expression or polynomial. They are variable, Coefficient and constant. In terms always the coefficient comes before the variable. The number part is called the coefficients. The variable part is the alphabetic part with one or more raised to a power. Polynomials can be classified based on number of variables, number of terms and degree.

FAQs

1. What is the leading coefficient in a polynomial?

In polynomials, the leading coefficient is the coefficient of the leading term or the term with highest degree. For example, 2x²+4x+3 here 2x² is the leading term and 2 is the leading coefficient.

2. What are the types of polynomials based on their terms ?

Polynomials are classified based on the number of terms, they are

• Monomial with one term Example 2x²

• Binomial with two terms Example 2x² + 2x

• Trinomial with three terms Example 2x²+ 3x+ 2

• Polynomial with many terms Example 2x⁴+ 3x³+ 35x + 35

3. How to divide two polynomials using the synthetic division method ?

To divide longer polynomials, we can perform both the long division method and synthetic division method. In the synthetic division method, we apply the divisor value equal to zero and substitute in the remainder obtained from division.

4. How does distributive law work in algebra ?

Distributive law is applicable for both addition and subtraction such as

$$\mathrm{a(b+c) = ab+ac\:\: and\:\: a(b-c) = ab -ac}$$

But distributive law is not applicable in both multiplication and division.

5. What are the conditions for algebraic expressions to be polynomial ?

For an algebraic expression to be polynomial, it must satisfy the following condition.

• All the powers in variables must be positive. Example 2x²+ 3x+ 2.

• The powers in any variable should not be in a fraction.

6. What are the possible conditions of zero of a polynomial?

Zero of polynomials have these conditions

• A zero of a polynomial need not to be zero.

• Maybe zero is the zero of a polynomial.

• Every linear polynomial has only one zero.

• A polynomial has one or more zero.