Polynomials

Lesson Plan - Get It!

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Before you can begin to calculate polynomials, you must first learn (or remember) what they are, so let's run through some definitions!

Monomials are numbers, variables, or a product of numbers and variables, with whole-number exponents.

For example, x², 5, and –7xy are monomials, because each example has a whole number exponent (numbers have an exponent of 0 unless stated otherwise).

X^-2 is not a monomial, because X^-2 has an exponent of –2, and negatives are not whole number exponents. 4x - y is not a monomial because it has two variables with exponents that are not being multiplied together.

For a quick refresher, watch Algebra Basics: What Are Polynomials? - Math Antics by Math Antics. Take notes on any points you are having difficulty understanding:

When you examine monomials, you can also look at the degree of a monomial, that is simply the sum of the exponents for the variable. Remember, constants (numbers without variables) have a degree of 0. A variable without a number (x, y, or a, for example) has a degree of 1.

Let's look at three examples of degrees:

1. 2a³b⁴ has a degree of 7, which can be calculated by adding the exponents together: 3 + 4 = 7.
2. 4 has a degree of 0 because it is a number, and numbers have a degree of 0.
3. 8y has a degree of 1, because any variable without a number (in this case, y) is assumed to have an exponent of 1, and the number 8 has a degree of 0.

For more examples, visit Degree (of an Expression) at MathsIsFun.com.

Try out a few examples. Find the degree of each monomial.

Once you have checked your answers, move on to polynomials.

A polynomial is a monomial, or multiple monomials being added or subtracted.

To find the degree of a polynomial, look at the degree of the term with the greatest degree. This is easily done by finding the degree of each term, then comparing them.

Try some examples:

1. 4x - 18x¹⁵ — The degree of 4x is 1. The degree of -18x¹⁵ is 15, so the degree of the entire polynomial is 15, because 15 is bigger than 1.
2. 6x⁴ + 9x² - x + 3 — The degree of 6x⁴ is 4. The degree of 9x² is 2. The degree of -x is 1. The degree of 3 is 0. The largest degree is 4, so the polynomial is of degree 4.
3. x²y + xy + .75 — The degree of x²y is 3 because 2 + 1 = 3. The degree of xy is 2 because 1 + 1 = 2. The degree of .75 is 0 because .75 is a constant. The degree of the polynomial is 3 because that is the largest degree.

Try a few examples. Find the degree of each polynomial.

The terms of a polynomial can technically be written in any order; however, the most efficient way to write polynomials is with only one variable in standard form, which is in descending degrees. This means if the polynomial only has the variable x, we count down from the largest exponent to the smallest, and to the constants (Remember: constants are numbers without a variable).

For example:

If we have 20x - 4x³ + 2 - x², we would rewrite this in standard form as -4x³ - x² + 20x + 2. This also tells us that our polynomial has a degree of 3.

The –4 in the previous example is the leading coefficient. It is also the number attached to the monomial with the greatest degree, and is the first coefficient when written in standard form.

For example, in 6y⁵ + y³ + 4y, the 6 would be the leading coefficient. In x² + 3x - 6, the coefficient is 1. Remember, any variable without a number in front of it is assumed to have a coefficient of 1.

Try a few and share your answers with a teacher or parent:

Write each polynomial in standard form, then give the leading coefficient of each polynomial.

Polynomials are classified with special names based on their degree and number of terms. Sadly, there is no easy way to remember these, so make sure you practice each name! The highlighted terms are the most common:

Let's do a few examples classifying polynomials. Then, you will practice a few on your own:

Classify each polynomial below:

1. 5x - 6
   • Degree: 1
   • Number of terms: 2
   • 5x - 6 is a linear binomial
2. y²+ y + 4
   • Degree: 2
   • Number of terms: 3
   • y² + y + 4 is a quadratic trinomial
3. 6x⁷ + 9x²- x + 3
   • Degree: 7
   • Number of terms: 4
   • 6x⁷ + 9x² - x + 3 is a 7^th-degree polynomial

Share your answers with a teacher or parent.

Classify each polynomial in terms of degrees and terms.

We have covered how to find the degree of a polynomial and monomial, how to write polynomials in standard form, and how to classify polynomials.

Now, continue on to the Got It? section for some polynomial practice!