What is a Polynomial Lesson

Before we get into the definition of a polynomial, it is important to understand the definition of a "term". A term is a number, variable, or the product of a number and one or more variables raised to powers. Terms are separated by "+" or "-" signs. In our above picture, we have the algebraic expression:
7x² + 6x - 9
The 7x², 6x, and 9 are terms. Notice how they are separated by "+" or "-" signs.
"Like Terms" occur when two or more terms have exactly the same variable parts. This means the variable(s) is(are) the same and raised to the same power(s).
5x², 9x² » Like Terms: each has the variable x raised to the 2nd power.
9x³y², 15x³y² » Like Terms: each has the variable x raised to the 3rd power and the variable y raised to the 2nd power.
2x⁴, 13x » Not Like Terms: each has the variable x, but our exponents are not the same.
11x³, 14y³ » Not Like Terms: although the exponents are the same, the variables are different.
We can add or subtract "like terms" by keeping the variable part the same and performing operations with the coefficients. We are just using the distributive property.
3x - 5x = (3 - 5)x = -2x
13x² + 9x² = (13 + 9)x² = 22x²

What is a Polynomial

A polynomial is either a single term or a finite sum of terms where all variables have whole number exponents and there are no variables in any denominator. A polynomial is the simplest type of algebraic expression.
Examples of Polynomials: $$11$$ $$-2x^3-5x$$ $$12x^2-7x+1$$ Examples of Non-Polynomials: $$\frac{2x - 5}{x^2}$$ $$5x^{-4}+ 1$$ $$\sqrt{7 - x}$$ Some polynomials occur so often, that we give them special names. A monomial is a polynomial with only one term.
Examples of Monomials:
8x⁷
12x⁴
-6x⁵y³
A binomial is a polynomial with two terms.
Examples of Binomials:
9x⁴ - 2
19x³ - 4x²
-11x⁵ + 7
A trinomial is a polynomial with three terms.
Examples of Trinomials:
25x³ + 2x - 14
-5x² - 12x + 1
3x⁹ + 4x⁵ + 6

Standard Form of a Polynomial

A polynomial is written in standard form when the powers are in descending order.
9x + 7x⁴ - 2 + 12x² - 3x³
We can rewrite this polynomial in standard form:
7x⁴ - 3x³ + 12x² + 9x - 2
Notice how the first or leftmost term "7x⁴" has the largest exponent on x. Then each position that follows has the next largest exponent on x. The final term (rightmost) is a 2. We can think about this as 2x⁰ since x⁰ is 1 and 2 • 1 is 2. Generally speaking, we are expected to write any polynomial answer in standard form.

The Degree of a Polynomial

The degree of a term in a polynomial is the sum of the exponents on the variable(s) of the term.
12x » The degree is 1
4x⁹ » The degree is 9
-10x⁵y³ » The degree is 8
7x³y⁷z¹⁴ » The degree is 24
4 » The degree is 0 (4x⁰ = 4 • 1 = 4)
The largest degree of any nonzero term in the polynomial is the degree of the polynomial.
7x² - x + 1 » degree of 2
x³y² + 4xy - 3 » degree of 5
x¹⁸y³z⁵ - 9x²y³z + 14 » degree of 26

Standard Form of a Polynomial with More Than One Variable

What happens when we want to write a polynomial with two variables in standard form? For example: $$2x^4y+3x^3y^3-5x^2y^3+10$$ In this case, we still list the term with the highest degree first, then the next highest, so on and so forth.
2x⁴y » degree of 5
3x³y³ » degree of 6
-5x²y³ » degree of 5
10 » degree of 0
So it's really straightforward that 3x³y³ goes first, but what happens here when we have a tie? This won't happen with one variable, because in that case, we would have like terms and be able to simplify. So here what we do is think about alphabetical order. Since x comes before y, we would think about which term has the x with the larger exponent. 2x⁴y has a larger exponent on x, so it will be listed first. $$3x^3y^3+2x^4y-5x^2y^3+10$$

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