Write and Classify
Polynomials in Standard Form
Jen Kershaw
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Printed: April 15, 2015
AUTHOR
Jen Kershaw
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C HAPTER
Chapter 1. Write and Classify Polynomials in Standard Form
1
Write and Classify
Polynomials in Standard Form
Here you’ll write and classify polynomials in standard form.
Do you know how to measure the degree of a polynomial? Take a look at this dilemma.
4x3 + 3x + 9
What if you had this polynomial? Can you identify it by degree? Is it in standard form?
This Concept will teach you all that you need to know to answer these questions.
Guidance
Sometimes, you will see an expression or an equation that has exponents and variables in it. These expressions and
equations can have more than one variable and sometimes more than one exponent in them. To understand how to
work with these variables and exponents, we have to understand polynomials.
A polynomial is an algebraic expression that shows the sum of monomials .
Polynomials: x2 + 5
3x − 8 + 4x5
− 7a2 + 9b − 4b3 + 6
We call an expression with a single term 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—“five-term polynomial.”
Did you know that you can write and classify polynomials?
First, let’s think about how we can classify each polynomial. We 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.
x2 has an exponent of 2, so it is a term to the second degree.
−2x5 has an exponent of 5, so it is a term to the fifth degree.
x2 y has an exponent of 2 on the x and an unwritten exponent of 1 on the y,
so this term is to the third degree (2 + 1). Notice that we add the two degrees together because it has two variables.
8 is a monomial that is a constant with no variable, its degree is zero.
We can also work on the ways that we write polynomials. One way to write a polynomial is in what we call
standard form.
In order to write any polynomial in standard form, we look at the degree of each term. We then write each
term in order of degree, from highest to lowest, left to write.
Take a look.
Write the expression 3x − 8 + 4x5 in standard form.
This is a trinomial. 3x has a degree of 1, -8 has a degree of zero, and 4x5 has a degree of 5. We write these in order
by degree, highest to lowest:
4x5 + 3x − 8
The degree of a polynomial is the same as the degree of the highest term, so this expression is called “a fifth-degree
1
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trinomial.”
Name the degree of each polynomial.
Example A
5x4 + 3x3 + 9x2
Solution: Fourth degree
Example B
6y3 + 3xy + 9
Solution: Third degree
Example C
7x2 + 3x + 9y
Solution: Second degree
Now let’s go back to the dilemma from the beginning of the Concept.
4x3 + 3x + 9
The highest exponent here is a 3, so we can say that this is a polynomial to the third degree. Since the values go
from the highest degree to the least degree, we can say that this polynomial is in standard form already.
Guided Practice
Here is one for you to try on your own.
Write the following polynomial in standard form.
4x3 + 3x5 + 9x4 − 2xy + 11
Solution
To accomplish this task, rewrite the polynomials so that the exponents are in descending order.
3x5 + 9x4 + 4x3 − 2xy + 11
This is our answer.
Video Review
MEDIA
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URL: http://www.ck12.org/flx/render/embeddedobject/60114
Introduction to Polynomials
2
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Chapter 1. Write and Classify Polynomials in Standard Form
Explore More
Directions: Write the following polynomials in standard form and then identify its degree:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
4x2 + 5x3 + x − 1
9 + 3y2 − 2y
8 + 3y3 + 8y + 9y2
y + 6y4 − 2y3 + y2
−16y6 − 18
3x + 2x2 + 9y + 8
8y4 + y − 7y3 − 3y2
−3 + 8x2 − 2x3 − x
9 − 3y2 − 2y3 + 2y
14 + 6x2 − 2x − 8y
4x + 3x2 − 5x3 + 8x4
−8 + 3y2 − 2y3 + y
9 + 8y2 + 2y3 − 8y
m4 − 12m7 + 6m5 − 6m − 8
−x3 y2 + 5x3 y + 8xy
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