Базові факти про многочлени

DEFINITIONS

A polynomial is a mathematical expression of one or more algebraic terms (monomials).

The word "polynomial" comes from the root words poly- (many) and -nomial (terms).

A polynomial can have constants (numbers), variables (letters), and exponents (powers), but never negative exponents or division by a variable.

Polynomials like to be written in standard form. This means they contain no like terms and the exponents are in descending order (largest to smallest).

The degree of a polynomial with only one variable is the largest exponent of that variable.

Sometimes we have to put a polynomial in standard form before finding the degree, in particular if the distributive property has to come into play.

Example: Find the degree of $x^2(5 + x) + x^3.$

Solution: We need to get rid of any parentheses because standard form doesn't use parentheses. Let's distribute the $x^2:$

$$x^2(5 + x) + x^3 = 5x^2 + x^3 + x^3.$$

Now combine like terms and arrange the terms to be in standard form:

$$5x^2 + x^3 + x^3 = 5x^2 + 2x^3 = 2x^3 + 5x^2.$$

The terms are now in decreasing order by degree, so the degree is $3,$ since the first term is an $x^3$ term.

CLASSIFYING POLYNOMIALS

Polynomials can be classified in two ways: by degree (highest exponent) and by their number of terms (terms are separated by addition or subtraction signs).

When classifying by number of terms, it can be helpful to think of different kinds of cycles: for instance, a monocycle has one wheel, a bicycle has two wheels, a tricycle has three wheels, et cetera.

When you've got multiple polynomials together in standard form, you can create a polynomial equation or polynomial function, where the higher degrees are at the start of the equation. You'll see this with quadratic equations, where the number of terms is up to $3$! You can also find polynomials in rational expressions.

(Videos) Basics of Polynomials

by mathman1024

This video by mathman1024 goes over the basics of polynomials.

$\large{\text{Summary}}$

A polynomial is the sum of terms. An example of a polynomial is: $$x^2+6x+7$$

The degree of a term is the sum of the exponents of the variable factors.

Lets look at an example with a larger polynomial. $$10x^{11}-7x^2y^5+8xy^7-120x^4+xy^2a-3$$ Let's find the degree of each term. $$\begin{array}{c|c} \text{Term}&\text{Degree}\\ \hline 10x^{11}&11\\ -7x^2y^5&7\\ 8xy^7&8\\ -120x^4&4\\ xy^2a&4\\ -3&0 \end{array}$$

The degree of a polynomial is the largest degree of its terms.

Looking at our last example, the degree of that polynomial would be $11$.

(Video) Intro to Polynomials

by mathman1024

This video goes over some more definitions and shows a quick example of adding and subtracting polynomials.

A monomial is a polynomial with only one term.

A binomial is a polynomial with only two terms.

A trinomial is a polynomial with only three terms.

To add and subtract polynomials, even ones with multiple variables, we just combine like terms.

Let's try it out with the example, $(3x^2+7x-10)+(5x^2-10x+3)$. $$\begin{align} &(3x^2+7x-10)+(5x^2-10x+3)\\ &=(3x^2+5x^2)+(7x-10x)+(-10+3)\\ &=8x^2-3x-7 \end{align}$$

Now let's try subtraction, $(-6x^2+4x-15)-(3x^2-7x+2)$. $$\begin{align} &(-6x^2+4x-15)-(3x^2-7x+2)\\ &=(-6x^2-3x^2)+(4x--7x)+(-15-2)\\ &=-9x^2+11x-17 \end{align}$$

What is a Polynomial?

A polynomial is a mathematical expression made up of two or more terms joined together by addition or subtraction.

These terms can be constants (numbers on their own, without variables), variables (like $x$ and $y$, sometimes with exponents), or variables with coefficients.

Let's take a look at some examples:

Let's see some examples:

$$9-4x\\ 3x^2+4x-5\\ 2x+7$$

Each of these expressions contains two or more terms joined by addition or subtraction, so they are all polynomials. One key thing to note about all of these examples is that each of the terms has a different exponent on the variable. For example, with the first example, $9-4x$, we see that the $9$ has no associated variable, and the $4x$ has a variable with an exponent of $1$.

Let's also examine the last polynomial example from the list above in more depth:

We can also get more specific in terms of how we name our polynomials. Take a look at this chart:

Consider the following expression: $x^4+\frac{5}{2}x-1$

Which of these terms BEST describes it? It has $3$ terms so it is a trinomial.

What is the "Degree" of a Polynomial?

The degree of a polynomial can be determined by looking at the exponents of its terms. Here are the general guidelines to follow when figuring out a polynomial's degree:

1. The degree of the polynomial is the same as the highest exponent it contains.
2. If you have a term with more than one exponent, the degree is the sum of those exponents.
3. Arrange your polynomials so that you start with the term of the highest degree and end with the term of the lowest degree.

Let's look at each of these guidelines in action.

1. The degree of the polynomial is the same as the highest exponent it contains

2. If you have a term with more than one exponent, the degree is the sum of those exponents

3. Arrange your polynomials from highest to lowest degree terms

A polynomial is in standard form when the terms are ordered by highest to lowest degree.