Lesson Objectives
  • Learn how to determine if two or more terms are "like terms"
  • Learn the basic definition of a polynomial
  • Learn how to find the degree of a polynomial
  • Learn how to write a polynomial in standard form

What is a Polynomial?

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. What is a Term in Math, shows three terms separated by '+' and '-' 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:
A binomial is a polynomial with two terms.
Examples of Binomials:
9x4 - 2
19x3 - 4x2
-11x5 + 7
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 non-zero 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 straight forward 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. $$3x^3y^3+2x^4y-5x^2y^3+10$$

Skills Check:

Example #1

Determine which pair of terms are "not like terms"

Please choose the best answer.

$$3x^2, 5x^2$$
$$7xyz, -2zxy$$
$$5x^2y^2z, -4zx^2y^2$$
$$6x^2yz, 5y^2xz$$
$$-x^2y^4, 15y^4x^2$$

Example #2

Determine which is "not a polynomial"

Please choose the best answer.

$$4x^{-2}- 8x - 1$$
$$-x^2 + x - 7$$
$$\frac{3x^2 - 5}{2}$$
$$\sqrt{3}\cdot x^3 - 8$$

Example #3

Find the degree of the polynomial $$5x^2y^9 + 7xy^{15}- 12x^{19}$$

Please choose the best answer.


Example #4

Write in standard form $$3x^5 + 9x^9 - 4 + 11x$$

Please choose the best answer.

$$3x^5 + 9x^9 - 4 + 11x$$
$$11x + 3x^5 + 9x^9 + 4$$
$$9x^9 - 4 + 11x + 3x^5$$
$$9x^9 + 3x^5 + 11x - 4$$
$$9x^9 + 11x - 3x^5 - 4$$
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