## Factors And Coefficients Of A Polynomial

**Factor: **

When numbers (constants) and variables are multiplied to form a term, then each quantity multiplied is called a **factor** of the term. A constant factor is called a numerical factor while a variable factor is called a literal factor.

**For Example:**

(i) 7, x and 7x are factors of 7x, in which

7 is constant (numerical) factor and x is variable (literal) factor.

(ii) In 5x^{2}y, the numerical factor is –5 and literal factors are : x, y, xy, x^{2} and x^{2}y.

**Coefficient: **

Any factor of a term is called the **coefficient** of the product of the remaining factors.

There are two types of coefficients:

1. Numerical coefficient or simply coefficient

2. Literal coefficient

**For Example:**

(i) In 7x ; 7 is coefficient of x

(ii) In 7xy, the numerical coefficient of the term 7xy is 7 and the literal coefficient is xy.

In a more general way,

Coefficient of xy = 7

Coefficient of 7x = y

Coefficient of 7y = x

(iii) In (- mn^{2}), the numerical coefficient of the term is (- 1) and the literal coefficient is mn^{2}.

In a more general way,

Coefficient of mn^{2} = – 1

Coefficient of (-n^{2}) = m

Coefficient of m = (- n^{2})

(iv) In –5x^{2}y; 5 is coefficient of –x^{2}y; –5 is coefficient of x^{2}y.

**Like and unlike terms:** Two or more terms having the same algebraic factors are called like terms, and two or more terms having different algebraic factors are called unlike terms.

**Example:** In the expression 5x^{2} + 7xy – 7y – 5xy, look at the terms 7xy and (- 5xy). The factors of 7xy are 7, x, and y and the factors of (- 5xy) are (- 5), x, and y. The algebraic factors (which contain variables) of both terms are x and y. Hence, they are like terms. Other terms 5x^{2} and (- 7y) have different algebraic factors [5 × x × x and (- 7y)]. Hence, they are unlike terms.

## Factors And Coefficients Of A Polynomial With Examples

**Example 1: **Write the coefficient of:

(i) x^{2} in 3x^{3} – 5x^{2} + 7

(ii) xy in 8xyz

(iii) –y in 2y^{2} – 6y + 2

(iv) x^{0} in 3x + 7

**Solution:**

(i) –5

(ii) 8z

(iii) 6

(iv) Since x^{0} = 1,

Therefore 3x + 7 = 3x + 7x^{0}

coefficient of x^{0} is 7.

**Example 2: ** Find the terms and factors of algebraic expression 8x^{2} – 3x.

**Solution:**

**Example 3: ** Find the terms and factors of algebraic expression 5x^{3} + 7xy – y^{2}.

**Solution:**

This is called the tree diagram and it is the best way to represent expression, terms, and factors.

**Example 4:** Identify like terms in the following:

2xy, -xy^{2}, x^{2}y, 5y, 8yx, 12yx^{2}, -11xy

**Solution:** 2xy, 8yx, -11xy are like terms having the same algebraic factors x and y.

x^{2}y and 12yx^{2} are also like terms having the same algebraic factors x, x and y.

**Example 5:** State whether the given pairs of terms are like or unlike terms:

(a) 19x, 19y (b) 4m^{2}p, 7pm^{2}

**Solution:**

(a) 19x and 19y are unlike terms having different algebraic factors, i.e., x and y.

(b) 4m^{2}p, 7pm^{2} are like terms having the same algebraic factors, i.e., m, m, p.