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### Course: Integrated math 3>Unit 1

Lesson 1: Intro to polynomials

# Polynomials intro

This introduction to polynomials covers common terminology like terms, degree, standard form, monomial, binomial and trinomial. Polynomials are sums of terms of the form k⋅xⁿ, where k is any number and n is a positive integer. For example, 3x+2x-5 is a polynomial.  Created by 1. Hello Fren.

## Want to join the conversation?

• why terms with negetive exponent not consider as polynomial?
• It is because of what is accepted by the math world. If you have 5^-2, it can be simplified to 1/5^2 or 1/25; therefore, anything to the negative power isn't in its simplest form. Also, not sure if Sal goes over it but you can't have a term being divided by a variable for it to be a polynomial (ie 2/x+2) However, (6x+5x^2)/(x) is a polynomial because once simplified it becomes 6+5x or 5x+6.
• I have four terms in a problem is the problem considered a trinomial
• "mono" meaning one
"bi" meaning two
"tri" meaning three
and "poly" meaning "many"
so,
if you have one term its a monomial
if you have two terms its a binomial
if you have three terms its a trinomial
if you have a four terms its a four term polynomial
if you have more than four terms then for example five terms you will have a five term polynomial and so on

hope this helped!
• When we write a polynomial in standard form, the highest-degree term comes first, right? And leading coefficients are the coefficients of the first term. So does that also mean that leading coefficients are the coefficients of the highest-degree terms of any polynomial, regardless of their order?
• Correct, standard form means that the terms are ordered from biggest exponent to lowest exponent. The leading coefficient is the coefficient of the first term in a polynomial in standard form. For example, 3x^4 + x^3 - 2x^2 + 7x. This polynomial is in standard form, and the leading coefficient is 3, because it is the coefficient of the first term. Keep in mind that for any polynomial, there is only one leading coefficient.
• Can x be a polynomial term?
• Yes, "x" can be a polynomial term. It can even be a polynomials called a monomial.
• A constant has what degree?
• A constant would be to the 0th degree while a linear is to the 1st power, quadratic is to the 2nd, cubic is to the 3rd, the quartic is to the 4th, the quintic is to the fifth, and any degree that is 6 or over 6 then you would say 'to the __ degree, or of the __ degree.
• If I have something like (2x+3)(5x+4) would this be a binomial if not what can I call it?
• It is the multiplication of two binomials which would create a trinomial if you double distributed (10x^2 +23x + 12).
• So the term 6 by itself is a monomial and a polynomial? I'm confused, can someone explain this a bit clearer?
• term 6 is a polynomial. it is also a mononomial which is a subclassification.
• I now know how to identify polynomial. But how do you identify trinomial, Monomials, and Binomials
• They are all polynomials.
Sal goes thru their definitions starting at in the video.
A monomials is a polynomial with only 1 term
A binomial is a polynomial with 2 terms.
A trinomial is a polynomial with 3 terms.
Hope this helps.
• I have a few doubts...
Why should a polynomial have only non-negative integer powers, why not negative numbers and fractions?Is there any specific name for those expressions with a variable as a power and why can't such expressions be polynomials?
• Given that x^-1 = 1/x, a polynomial that contains negative exponents would have a variable in the denominator. This drastically changes the shape of the graph, adding values at which the graph is undefined and changes the shape of the curve since a variable in the denominator behaves differently than variables in the numerator would. These are called rational functions. Equations with variables as powers are called exponential functions. They are curves that have a constantly increasing slope and an asymptote.
• Is k⋅xⁿ a polynomial if k is an irrational number? For example 5√-1⋅xⁿ, would this be considered a monomial?
• You have an exponential expression because "x" has a variable for its exponent. This indicates that the exponent can be any real number. With a polynomial, the exponents on variables must be whole number. So, your expression is not a polynomial.

5√(-1) is an imaginary number, not an irrational number. Irrational numbers would be a non-repeating and non-terminating decimal. Numbers like Pi, or a square root that contains a real number that is not a perfect square would be irrational numbers.

Now, if you had an expression: 5√(2) x^2, it would be a polynomial with a coefficient that is an irrational number.

The coefficients and constant terms in a polynomial are real numbers: integers, decimals, fractions, and irrational numbers. You would not have a complex number as a coefficient. Though, you can have solutions to polynomial equations that are complex numbers.

Hope this helps.