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In this section students will:
  • Identify the degree and leading coefficient of polynomials.
  • Add and subtract polynomials.
  • Multiply polynomials.
  • Use FOIL to multiply binomials.
  • Perform operations with polynomials of several variables.

Earl is building a doghouse, whose front is in the shape of a square topped with a triangle. There will be a rectangular door through which the dog can enter and exit the house. Earl wants to find the area of the front of the doghouse so that he can purchase the correct amount of paint. Using the measurements of the front of the house, shown in [link] , we can create an expression that combines several variable terms, allowing us to solve this problem and others like it.

Sketch of a house formed by a square and a triangle based on the top of the square. A rectangle is placed at the bottom center of the square to mark a doorway. The height of the door is labeled: x and the width of the door is labeled: 1 foot. The side of the square is labeled: 2x. The height of the triangle is labeled: 3/2 feet.

First find the area of the square in square feet.

A = s 2 = ( 2 x ) 2 = 4 x 2

Then find the area of the triangle in square feet.

A = 1 2 b h =    1 2 ( 2 x ) ( 3 2 ) =    3 2 x

Next find the area of the rectangular door in square feet.

A = l w = x 1 = x

The area of the front of the doghouse can be found by adding the areas of the square and the triangle, and then subtracting the area of the rectangle. When we do this, we get 4 x 2 + 3 2 x x ft 2 , or 4 x 2 + 1 2 x ft 2 .

In this section, we will examine expressions such as this one, which combine several variable terms.

Identifying the degree and leading coefficient of polynomials

The formula just found is an example of a polynomial    , which is a sum of or difference of terms, each consisting of a variable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as 384 π , is known as a coefficient    . Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or fractions. Each product a i x i , such as 384 π w , is a term of a polynomial    . If a term does not contain a variable, it is called a constant .

A polynomial containing only one term, such as 5 x 4 , is called a monomial    . A polynomial containing two terms, such as 2 x 9 , is called a binomial    . A polynomial containing three terms, such as −3 x 2 + 8 x 7 , is called a trinomial    .

We can find the degree    of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the leading term    because it is usually written first. The coefficient of the leading term is called the leading coefficient    . When a polynomial is written so that the powers are descending, we say that it is in standard form.

A polynomial reading: a sub n times x to the nth power plus and so on plus a sub 2 times x squared plus a sub one times x plus a subzero is shown. The a in the term a sub n is labeled: leading coefficient. The n in the term x to the nth power is labeled: degree. Finally, the entire term is labeled as: Leading term.

Polynomials

A polynomial    is an expression that can be written in the form

a n x n + ... + a 2 x 2 + a 1 x + a 0

Each real number a i is called a coefficient    . The number a 0 that is not multiplied by a variable is called a constant . Each product a i x i is a term of a polynomial    . The highest power of the variable that occurs in the polynomial is called the degree    of a polynomial. The leading term    is the term with the highest power, and its coefficient is called the leading coefficient    .

Given a polynomial expression, identify the degree and leading coefficient .

  1. Find the highest power of x to determine the degree.
  2. Identify the term containing the highest power of x to find the leading term.
  3. Identify the coefficient of the leading term.

Questions & Answers

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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