POLYNOMIAL

Polynomial

Definition Of Polynomial

A polynomial is a monomial or a sum or difference of two or more monomials.

More About Polynomial

Polynomial function: A polynomial function is a function whose terms each contain a constant multiplied by a power of a variable.
P(x) = a₀xⁿ + a₁x^n-1 + .................+ a_n-2x² + a_n-1x + an is a polynomial function
where a₀, a₁, a₂,..., a_n are all real numbers and n is a non-negative integer.
Polynomial in one variable: a₀xⁿ + a₁x^n-1 + .................+ a_n-2x² + a_n-1x + an is a polynomial in one variable.

Video Examples: Adding and Subtracting Polynomials

Example of Polynomial

The following are few examples of polynomials.
2x - 7, 5x² + 2x - 8, - 7y² + 4y - 13, 4x² + 16xy + 16y²

Solved Example on Polynomial

Ques: Simplify the polynomial expression: (10x³ + 12x + 15) - (3x³ + 6x + 10)

Choices:

A. 7x³ + 18x + 25
B. 7x³ + 6x + 5
C. 7x⁶ + 18x² + 25
D. 7x⁶ + 6x² + 5
Correct Answer: B

Solution:

Step 1: The given polynomial expression (10x³ + 12x + 15) - (3x³ + 6x + 10) can be simplified as 7x³ + 6x + 5. [Combine like terms.]

Quick Summary

Polynomials are expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication.
The exponents of variables in a polynomial must be non-negative integers.
Polynomials can be classified by the number of terms (monomial, binomial, trinomial) and by their degree (highest exponent).

\[ P(x) = a_{0}x^{n} + a_{1}x^{n-1} + \dots + a_{n-2}x^{2} + a_{n-1}x + a_{n} \]

🍎 Teacher Insights

Use visual aids and manipulatives to help students understand the concept of combining like terms. Emphasize the importance of careful organization and attention to detail when simplifying polynomial expressions.

🎓 Prerequisites

Variables
Exponents
Algebraic Expressions
Real Numbers

Check Your Knowledge

Q1: Simplify: (2x^2 + 3x - 1) + (x^2 - x + 4)

A. 3x^2 + 2x + 3 B. x^2 + 4x + 5 C. 2x^4 + 3x^2 + 3 D. 3x^2 + 4x + 3

Q2: Simplify: (10x^3 + 12x + 15) - (3x^3 + 6x + 10)

A. 7x^3 + 18x + 25 B. 7x^3 + 6x + 5 C. 7x^6 + 18x^2 + 25 D. 7x^6 + 6x^2 + 5

Frequently Asked Questions

Q: What is the degree of a polynomial?
A: The degree of a polynomial is the highest power of the variable in the polynomial.

Q: Is x^(1/2) a polynomial?
A: No, because the exponent 1/2 is not a non-negative integer.