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.
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For Example:(i) 7, x and 7x are factors of 7x, in which7 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.
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 = 7Coefficient of 7x = yCoefficient 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 = – 1Coefficient of (-n2) = mCoefficient 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 + 7Solution:(i) –5(ii) 8z(iii) 6(iv) Since x0 = 1,Therefore 3x + 7 = 3x + 7x0coefficient of x0 is 7.
Example 2: Identify like terms in the following:2xy, -xy2, x2y, 5y, 8yx, 12yx2, -11xySolution: 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.
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Example 3: State whether the given pairs of terms are like or unlike terms:(a) 19x, 19y (b) 4m2p, 7pm2Solution:(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.
Filed Under: Mathematics Tagged With: Coefficients, Factors, Factors And Coefficients Examples, Like and unlike terms, Polynomials