Polynomial Definitions, Naming and Terms
Description/Explanation/Highlights
Video Description
This video explains what a polynomial is, how they are named and what a term is in a polynomial.
Steps and Key Points to Remember
To determine if an expression is a polynomial, follow these steps:
- The word polynomial means “many terms” so first begin by finding the term or terms that make up the possible polynomial.
- Terms are the parts of the expression separated by “+” and “-” signs.
- A polynomial is made up of one or more terms that fit a set of rules. For example, \(3x^2+2x+5\) is a polynomial of three terms: \(3x^2\), \(2x\), and \(5\).
- To see if an expression is a polynomial, look at each term and make sure that: there are no negative exponents on a variable, there are no variables in a denominator, and there are no fractional exponents.
- \(4x^3+3x^2+2xy+y^2\) is a 4-term polynomial since there are 4 terms without negative exponents, variables in denominators, or fractional exponents.
- \(3x^{-2}+4x^3-1\) is NOT a polynomial since the first term has a variable with a negative exponent.
- \(3x^2\) is a one-term polynomial since it meets the rules of a polynomial.
- \(\frac{3x^2}{y}\) is not a polynomial since y is in the denominator.
- \(5x^{\frac{1}{2}}+4x+1\) is not a polynomial since there is a fractional exponent in the first term of the expression.
- \(2\sqrt[3]{x}+4\) is NOT a polynomial because roots can be rewritten as fractional exponents which are not allowed. In this case the expression could be written as: \(2x^\frac{1}{3}+4\) which would have fractional exponent.
- \(4x^2y^2\) is a 1-term polynomial called a monomial.
- \(4x^2y^2-2x^2y\) is a 2-term polynomial called a binomial.
- \(3x^2+2x+4\) is a 3-term polynomial called a trinomial.
- \(3x^3+2x^2+5x-2\) is a 4-term polynomial.
Here are some key points to keep in mind when identifying polynomials.
- Polynomials are expressions made up of terms separated by “+” and “-” signs.
- To be classified as a polynomial, no term can have a variable with a negative exponent, a variable in the denominator, or a fractional exponent.
- Remember that in determining whether an expression is a polynomial, roots also prevent an expression from being classified as a polynomial since they can be rewritten as fractional exponents.
- Polynomials with one term are classified as monomials.
- Polynomials with two terms are called binomials.
- Polynomials with three terms are called trinomials.
- Polynomials with four or more terms are generally just called polynomials.
Video Highlights
- 00:00 Introduction
- 00:20 Definition of a polynomial
- 00:35 Definition of terms
- 00:50 \(3x^2+2x+5\) example of a polynomial
- 01:20 Rules for what terms in a polynomial cannot contain
- 02:55 \(4x^3+3x^2+2xy+y^2\) 4-term polynomial example
- 03:30 \(3x^{-2}+4x^3-1\) example of an expression with a negative exponent that is not a polynomial
- 03:50 \(3x^2\) example of a monomial
- 04:00 \(\frac{3x^2}{y}\) example of a single-term expression that is not a monomial because it has a variable in the denominator
- 00:50 \(3x^2+2x+5\) example of a polynomial
- 04:12 \(5x^{\frac{1}{2}}+4x+1\) example of a three-term expression that is not a polynomial because of a fractional exponent
- 04:25 \(2\sqrt[3]{x}+4\) example of an expression that is not a polynomial because of the cube root that could be rewritten as a fractional exponent
- 04:55 \(4x^2y^2\) example of a monomial
- 05:25 \(4x^2y^2-2x^2y\) example of a binomial
- 05:47 \(3x^2+2x+4\) example of a trinomial
- 06:07 \(3x^3+2x^2+5x-2\) example of a 4-term polynomial
- 06:55 Conclusion
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