Polynomials are mathematical expressions made up of variables, coefficients, and exponentiation. They can have one or more terms, where each term consists of a coefficient multiplied by one or more variables raised to non-negative integer exponents.
Here’s a general form of a polynomial:
In this expression:
- P(x) represents the polynomial function.
- x is the variable.
- n is a non-negative integer and represents the degree of the polynomial.
- an,an−1,…,a0 are coefficients, which can be real numbers, integers, or other algebraic expressions.
Here are some basic operations and concepts related to polynomials:
- Addition and Subtraction: You can add and subtract polynomials by combining like terms. Like terms are terms with the same variables raised to the same powers. For example, 3x2 and 5x2 are like terms.
- Multiplication: To multiply two polynomials, use the distributive property (FOIL method for binomials – First, Outer, Inner, Last) to multiply each term of one polynomial by each term of the other polynomial and then combine like terms.
- Division: Division of polynomials can be more complex, involving long division or synthetic division. You divide the leading term of the dividend by the leading term of the divisor to find the first term of the quotient, then subtract and continue the process.
- Degree: The highest exponent of the variable in a polynomial term determines the degree of that term. The degree of the polynomial is the highest degree among its terms.
- Zeroes or Roots: The values of x that make the polynomial equal to zero are called the zeroes or roots of the polynomial.
- Factoring: Factoring involves expressing a polynomial as a product of its irreducible factors. This is useful for simplifying expressions and solving equations.
- Synthetic Substitution: A method to evaluate a polynomial at a specific value of x without actually performing all the calculations.
- Long Division of Polynomials: A method used to divide one polynomial by another polynomial.
- Binomial Theorem: A formula that provides a way to expand powers of binomials without actually multiplying them out.
Remember that practice is key when working with polynomials or anything in math, for that matter…