Factors And Coefficients Of A Polynomial
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.
(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.
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
(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
(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.
Example 3: Find the terms and factors of algebraic expression 5x3 + 7xy – y2.
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
(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.