Math's: Algebra and Algebraic Expressions: Polynomials: Introduction, Degrees, Evaluation and Zeroes of Polynomials

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Polynomials (Introduction)

An algebraic expression, in which variable(s) does (do) not occur in the denominator, powers of variable(s) are whole numbers and numerical coefficients of various terms are real numbers, is called polynomial.

Example- is a polynomial.

  • A polynomial having only one term is known as monomial. Ex-

  • A polynomial that consists of two terms is known as binomial. Ex-

  • A polynomial consisting of three terms is known as trinomial. Ex-

  • The terms of polynomial having the same variable and the same power of the variable are known as like terms. Ex- In the expression, , and are like terms.

1. Which of the following algebraic expressions are polynomials?

(i)

(ii)

(iii)

(iv)

Solution: (i) is a polynomial as it satisfies all the conditions of polynomial.

(ii) is not a polynomial since one of the variable y occur in denominator

(iii) is a polynomial as it satisfies all the conditions of a polynomial.

(iv) is not a polynomial since one of the variable z has a fractional power.

2. Write the like terms in the equation .

Solution: Like Terms are as both the variables x and y and their power (which is one) are same.

Degree of Polynomials

The degree of a polynomial is the same as the degree of its term or terms having the highest degree and non-zero coefficient.

For Example – The degree of will be 4 as the highest power of a term is 4.

Similarly, in polynomial , the degree of polynomial is 5 since the highest degree is of the term .

  • A polynomial of degree 2 is also called a quadratic polynomial. For example, and are quadratic polynomials.

  • The degree of a non-zero constant polynomial is taken as zero.

  • When all the coefficients of variable(s) in the terms of a polynomial are zeros, the polynomial is called a zero polynomial. The degree of a zero polynomial is not defined.

Evaluation of Polynomials

Evaluation of polynomials means obtaining the numerical value of a polynomial substituting the value(s) of variable(s) in it.

Evaluation of Polynomials

Evaluation of Polynomials

For Ex- Evaluate for

Sol-

.

Zeroes of Polynomials

The value(s) of the variable for which the value of a polynomial in one variable is zero is (are) called zero(s) of the polynomial.

The value of the polynomial for is zero. Therefore, we say that is a zero of the polynomials .

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