Factors And Coefficients Of A Polynomial

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 5x²y, the numerical factor is –5 and literal factors are : x, y, xy, x² and x²y.

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 (- mn²), the numerical coefficient of the term is (- 1) and the literal coefficient is mn².

In a more general way,

Coefficient of mn² = – 1

Coefficient of (-n²) = m

Coefficient of m = (- n²)

(iv)  In –5x²y; 5 is coefficient of ­–x²y; –5 is coefficient of x²y.

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 5x² + 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 5x² 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)   x² in 3x³ – 5x² + 7

(ii)  xy in 8xyz

(iii) –y in 2y² – 6y + 2

(iv) x⁰ in 3x + 7

Solution:

(i) –5

(ii) 8z

(iii) 6

(iv) Since x⁰ = 1,

Therefore 3x + 7 = 3x + 7x⁰

coefficient of x⁰ is 7.

Example 2: Identify like terms in the following:

2xy, -xy², x²y, 5y, 8yx, 12yx², -11xy

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

x²y and 12yx² are also like terms having the same algebraic factors x, x and y.

Example 3: State whether the given pairs of terms are like or unlike terms:

(a) 19x, 19y (b) 4m²p, 7pm²

Solution:

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

(b) 4m²p, 7pm² are like terms having the same algebraic factors, i.e., m, m, p.