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Polynomials in Standard Form

Understand polynomials as specific kinds of Algebraic expressions

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Write and Classify Polynomials in Standard Form
License: CC BY-NC 3.0

Erin’s Math class was learning how to measure the degree of a polynomial. She was presented with the card shown above. What if you had this polynomial? Can you identify it by degree? Is it in standard form?

In this concept, you will learn to write and classify polynomials in standard form.

Polynomials

A polynomial is an algebraic expression that shows the sum of monomials.

Here are some polynomials.

\begin{align*}x^2+5 \quad \quad 3x-8+4x^5 \quad \quad -7a^2+9b-4b^3+6\end{align*}

An expression with a single term is a monomial, an expression with two terms is a binomial, and an expression with three terms is a trinomial. An expression with more than three terms is named simply by its number of terms.

First, let’s think about how you can classify each polynomial. You classify them according to terms. Each term can be classified by its degree.

The degree of a term is determined by the exponent of the variable or the sum of the exponents of the variables in that term.

The expression \begin{align*}x^2\end{align*} has an exponent of 2, so it is a term to the second degree.

The expression \begin{align*}-2x^5\end{align*} has an exponent of 5, so it is a term to the fifth degree.

The expression \begin{align*}x^2y\end{align*} has an exponent of 2 on the \begin{align*}x\end{align*} and an unwritten exponent of 1 on the \begin{align*}y\end{align*}, so this term is to the third degree \begin{align*}(2 + 1)\end{align*}. Notice that you add the two degrees together because it has two variables.

The expression 8 is a monomial that is a constant with no variable, its degree is zero.

You can also work on the ways that you write polynomials. One way to write a polynomial is in standard form. In order to write any polynomial in standard form, you look at the degree of each term. You then write each term in order of degree, from highest to lowest, left to write.

Let’s look at an example.

Write the expression \begin{align*}3x-8+4x^5\end{align*} in standard form.

First, look at the degrees for each term in the expression.

\begin{align*}3x\end{align*} has a degree of 1

8 has a degree of 0

\begin{align*}4x^5\end{align*} has a degree of 5

Next, write this trinomial in order by degree, highest to lowest

\begin{align*}4x^5+3x-8\end{align*}

The answer is \begin{align*}4x^5+3x-8\end{align*}.

The degree of a polynomial is the same as the degree of the highest term, so this expression is called a fifth degree trinomial.

Examples

Example 1

Earlier, you were given a problem about Erin and the polynomial.

Erin has to identify the degree of the polynomial \begin{align*}4x^3+3x+9\end{align*}.

First, look at the degrees for each term in the expression.

\begin{align*}4x^3\end{align*} has a degree of 3

\begin{align*}3x\end{align*} has a degree of 1

\begin{align*}9\end{align*} has a degree of 0

Next, the highest degree identifies the degree of the polynomial.

The term \begin{align*}4x^3\end{align*} is the highest degree so the degree of the polynomial is 3.

The answer is that the polynomial is of the third degree.

Example 2

Write the following polynomial in standard form.

\begin{align*}4x^3+3x^5+9x^4-2xy+11\end{align*}

First, look at the degrees for each term in the expression.

\begin{align*}4x^3\end{align*} has a degree of 3

\begin{align*}3x^5\end{align*} has a degree of 5

\begin{align*}9x^4\end{align*} has a degree of 4

\begin{align*}-2xy\end{align*} has a degree of 2

11 has a degree of 0

Next, write this polynomial in order by degree, highest to lowest

\begin{align*}3x^5+9x^4+4x^3-2xy+11\end{align*}

The answer is \begin{align*}3x^5+9x^4+4x^3-2xy+11\end{align*}.

Example 3

Name the degree of the expression \begin{align*}5x^4+3x^3+9x^2\end{align*}.

First, look at the degrees for each term in the expression.

\begin{align*}5x^4\end{align*} has a degree of 4

\begin{align*}3x^3\end{align*} has a degree of 3

\begin{align*}9x^2\end{align*} has a degree of 2

Next, the highest degree identifies the degree of the polynomial.

The term \begin{align*}5x^4\end{align*} is the highest degree so the degree of the polynomial is 4.

The answer is that the polynomial is of the fourth degree.

Example 4

Name the degree of the expression \begin{align*}6y^3+3xy+9\end{align*}.

First, look at the degrees for each term in the expression.

\begin{align*}6y^3\end{align*} has a degree of 3

\begin{align*}3xy\end{align*} has a degree of 2

\begin{align*}9\end{align*} has a degree of 0

Next, the highest degree identifies the degree of the polynomial.

The term \begin{align*}6y^3\end{align*} is the highest degree so the degree of the polynomial is 3.

The answer is that the polynomial is of the third degree.

Example 5

Write the following polynomial in standard form and identify the degree of the polynomial.

\begin{align*}7x^2+3x-2x^4+8x^6-7\end{align*}

First, look at the degrees for each term in the expression.

\begin{align*}7x^2\end{align*} has a degree of 2

\begin{align*}3x\end{align*} has a degree of 1

\begin{align*}-2x^4\end{align*} has a degree of 4

\begin{align*}8x^6\end{align*} has a degree of 6

\begin{align*}-7\end{align*} has a degree of 0

Next, write this polynomial in order by degree, highest to lowest

\begin{align*}8x^6-2x^4+7x^2+3x-7\end{align*}

Then, the highest degree identifies the degree of the polynomial.

The term \begin{align*}8x^6\end{align*} is the highest degree so the degree of the polynomial is 6.

The answer is \begin{align*}8x^6-2x^4+7x^2+3x-7\end{align*} and the polynomial is of the sixth degree.

Review

Write the following polynomials in standard form and then identify its degree:

1. \begin{align*}4x^2+5x^3+x-1\end{align*}

2. \begin{align*}9+3y^2-2y\end{align*}

3. \begin{align*}8+3y^3+8y+9y^2\end{align*}

4. \begin{align*} y+6y^4-2y^3+y^2\end{align*}

5. \begin{align*}-16y^6-18\end{align*}

6. \begin{align*}3x+2x^2+9y+8\end{align*}

7. \begin{align*}8y^4+y-7y^3-3y^2\end{align*}

8. \begin{align*}-3+8x^2-2x^3-x\end{align*}

9. \begin{align*}9-3y^2-2y^3+2y\end{align*}

10. \begin{align*}14+6x^2-2x-8y\end{align*}

11. \begin{align*}4x+3x^2-5x^3+8x^4\end{align*}

12. \begin{align*}-8+3y^2-2y^3+y\end{align*}

13. \begin{align*}9+8y^2+2y^3-8y\end{align*}

14. \begin{align*}m^4-12m^7+6m^5-6m-8\end{align*}

15. \begin{align*}-x^3y^2+5x^3y+8xy\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 12.2. 

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Vocabulary

Binomial

A binomial is an expression with two terms. The prefix 'bi' means 'two'.

Coefficient

A coefficient is the number in front of a variable.

constant

A constant is a value that does not change. In Algebra, this is a number such as 3, 12, 342, etc., as opposed to a variable such as x, y or a.

Monomial

A monomial is an expression made up of only one term.

Polynomial

A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.

Polynomial Function

A polynomial function is a function defined by an expression with at least one algebraic term.

Trinomial

A trinomial is a mathematical expression with three terms.

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