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# Polynomials in Standard Form

## Understand polynomials as specific kinds of Algebraic expressions

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Practice Polynomials in Standard Form
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Write and Classify Polynomials in Standard Form

Credit: Brenda Meery
Source: CK-12 Foundation

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.

### Guidance

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

Here are some polynomials.

x2+53x8+4x57a2+9b4b3+6\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 x2\begin{align*}x^2\end{align*} has an exponent of 2, so it is a term to the second degree.

The expression 2x5\begin{align*}-2x^5\end{align*} has an exponent of 5, so it is a term to the fifth degree.

The expression x2y\begin{align*}x^2y\end{align*} has an exponent of 2 on the x\begin{align*}x\end{align*} and an unwritten exponent of 1 on the y\begin{align*}y\end{align*}, so this term is to the third degree (2+1)\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 3x8+4x5\begin{align*}3x-8+4x^5\end{align*} in standard form.

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

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

8 has a degree of 0

4x5\begin{align*}4x^5\end{align*} has a degree of 5

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

4x5+3x8\begin{align*}4x^5+3x-8\end{align*}

The answer is 4x5+3x8\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.

### Guided Practice

Write the following polynomial in standard form.

4x3+3x5+9x42xy+11\begin{align*}4x^3+3x^5+9x^4-2xy+11\end{align*}

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

4x3\begin{align*}4x^3\end{align*} has a degree of 3

3x5\begin{align*}3x^5\end{align*} has a degree of 5

9x4\begin{align*}9x^4\end{align*} has a degree of 4

2xy\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

3x5+9x4+4x32xy+11\begin{align*}3x^5+9x^4+4x^3-2xy+11\end{align*}

The answer is 3x5+9x4+4x32xy+11\begin{align*}3x^5+9x^4+4x^3-2xy+11\end{align*}.

### Examples

#### Example 1

Name the degree of the expression 5x4+3x3+9x2\begin{align*}5x^4+3x^3+9x^2\end{align*}.

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

5x4\begin{align*}5x^4\end{align*} has a degree of 4

3x3\begin{align*}3x^3\end{align*} has a degree of 3

9x2\begin{align*}9x^2\end{align*} has a degree of 2

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

The term 5x4\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 2

Name the degree of the expression 6y3+3xy+9\begin{align*}6y^3+3xy+9\end{align*}.

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

6y3\begin{align*}6y^3\end{align*} has a degree of 3

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

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

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

The term 6y3\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 3

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

7x2+3x2x4+8x67\begin{align*}7x^2+3x-2x^4+8x^6-7\end{align*}

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

7x2\begin{align*}7x^2\end{align*} has a degree of 2

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

2x4\begin{align*}-2x^4\end{align*} has a degree of 4

8x6\begin{align*}8x^6\end{align*} has a degree of 6

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

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

8x62x4+7x2+3x7\begin{align*}8x^6-2x^4+7x^2+3x-7\end{align*}

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

The term 8x6\begin{align*}8x^6\end{align*} is the highest degree so the degree of the polynomial is 6.

The answer is 8x62x4+7x2+3x7\begin{align*}8x^6-2x^4+7x^2+3x-7\end{align*} and the polynomial is of the sixth degree.

Credit: Eljay
Source: https://www.flickr.com/photos/eljay/3079772444/in/photolist-5G9D3U-aM7ZiZ-nS3oxd-aM7ZDF-3KEtA3-943Tpe-2AJ2T-7NzNjC-aM7VCx-nNzrws-aM81ut-aM81hR-6j5yG9-6j5xsu-bdWeW2-7NvWoz-bmpoit-aM81ba-7NvBtg-7NvHUK-7NvQZx-7NvCi4-efj1Ri-7NvTgF-2z7sHm-7NvwNx-7NzMJQ-7NztTC-5XFvgG-9QBbkU-7Nvwh6-aM7Ytv-aM7Yov-7F4Zu-aM8dmX-aM8dia-aM8dct-aM8d7i-aM8d1r-7NvLtp-472ff5-6YZdgx-7NvVKa-9mAXgJ-7NvW4K-7NvUsx-7NvUev-7NzTrs-7NvTCK-7NzSoY

Remember Erin and the polynomial?

Erin has to identify the degree of the polynomial 4x3+3x+9\begin{align*}4x^3+3x+9\end{align*}.

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

4x3\begin{align*}4x^3\end{align*} has a degree of 3

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

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

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

The term 4x3\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.

### Explore More

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

1. 4x2+5x3+x1\begin{align*}4x^2+5x^3+x-1\end{align*}

2. 9+3y22y\begin{align*}9+3y^2-2y\end{align*}

3. 8+3y3+8y+9y2\begin{align*}8+3y^3+8y+9y^2\end{align*}

4. y+6y42y3+y2\begin{align*} y+6y^4-2y^3+y^2\end{align*}

5. 16y618\begin{align*}-16y^6-18\end{align*}

6. 3x+2x2+9y+8\begin{align*}3x+2x^2+9y+8\end{align*}

7. 8y4+y7y33y2\begin{align*}8y^4+y-7y^3-3y^2\end{align*}

8. 3+8x22x3x\begin{align*}-3+8x^2-2x^3-x\end{align*}

9. 93y22y3+2y\begin{align*}9-3y^2-2y^3+2y\end{align*}

10. 14+6x22x8y\begin{align*}14+6x^2-2x-8y\end{align*}

11. 4x+3x25x3+8x4\begin{align*}4x+3x^2-5x^3+8x^4\end{align*}

12. 8+3y22y3+y\begin{align*}-8+3y^2-2y^3+y\end{align*}

13. 9+8y2+2y38y\begin{align*}9+8y^2+2y^3-8y\end{align*}

14. m412m7+6m56m8\begin{align*}m^4-12m^7+6m^5-6m-8\end{align*}

15. x3y2+5x3y+8xy\begin{align*}-x^3y^2+5x^3y+8xy\end{align*}

### Vocabulary Language: English

Binomial

Binomial

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

Coefficient

A coefficient is the number in front of a variable.
constant

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

Monomial

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

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

Polynomial Function

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

Trinomial

A trinomial is a mathematical expression with three terms.