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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

what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
Wrong question
Saad
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl 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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