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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

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana 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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