OP - Polynomials Lesson

Polynomials

Let's do a quick review on what polynomials are and the types of polynomials. A polynomial function is a function of the form:  $f\left(x\right)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ .

The exponents of the variables must be positive integers to be a polynomial. The degree of a polynomial is the term with the greatest exponent.  $a_0,\:a_1,\:...,\:a_n$  are real numbers and are called the coefficients of the polynomial. Since an expression with the variable in the denominator has a negative exponent it would not be a polynomial. The domain of a polynomial function is the set of real numbers.

To find the degree of the polynomial, you must find the degree of each monomial term (terms are separated by addition and or subtraction signs). In other words, add up the exponents of each term. The degree is determined by the exponent or sum of exponents that has the greatest value within the polynomial.

A monomial is an algebraic expression that is a constant, a variable, or a product of a constant and one or more variables. It is also called a constant because the degree is 0. Every number by itself is a constant. Polynomials literally means "many terms." Terms are separated by addition and or subtraction signs.

Here are some examples of various polynomials:

• Monomials ("mono" meaning one): 2, 3x, 5xy, 4x², and  $\frac{x}{3}$
• Binomials ("bi" meaning two): 2 + 3x; a + 2bc;  $\frac{7}{9}t^3+5w^2$
• Trinomials ("tri meaning three): $r^3-2+3x;\:t^3-a+2bc;\:\:\frac{7}{9}t^3+5w^2-11$

**There are no special names for polynomials with more than 3 terms.

We not only classify polynomials by the number of terms, we also classify by the degree.

Polynomial	Example	Degree
Constant	1	0
Linear	2x + 1	1
Quadratic	3x2  + 2x + 1	2
Cubic	4x3  + 3x2  + 2x + 1	3
Quartic	5x4  + 4x3  + 3x2  + 2x + 1	4

When polynomials are classified by their degree, the word "degree" is defined as the following:  The degree of a polynomial is the highest degree for a term. The degree of a term is the sum of the powers of each variable in the term.

Lastly, let's review what the definition of the term  "leading coefficient"  is before we move on. It is the first coefficient when the polynomial is written in its general or standard form. In the equation below. The leading coefficient is 8 and the degree is 9:  $8x^9+3x^5+2x^3+16$ .

Now watch the teaching videos on Polynomial Expressions and Rational Functions.

Now it is time for you to complete the Identifying & Classifying Polynomials Self-Assessment. This will allow you to practice recognizing polynomial functions, their classification by the number of terms and by degree, and the leading coefficients of each.

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