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:    Find the terms and factors of algebraic expression 8x² – 3x.
Solution:

Example 3:    Find the terms and factors of algebraic expression 5x³ + 7xy – y².
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, -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 5: 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.