7.1 An Introduction to Polynomials

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7.1 An Introduction to Polynomials

• Objectives Identify, evaluate, add, and
• subtract polynomials. Classify polynomials,
  and describe the shapes of their graphs.
• Standard 2.8.11.S. Analyze properties and
  relationships of polynomials.

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A monomial is a numeral, a variable, or the
product of a numeral and one or more variables. A
monomial with no variables, such as 1, is called
a constant. A coefficient is the numerical factor
in a monomial. The degree of a monomial is the
sum of the exponents of its variables. A
polynomial is a monomial or a sum of terms that
are monomials. A polynomial with two terms is a
binomial. A polynomial with three terms is a
trinomial. The degree of a polynomial is the same
as that of its term with the greatest degree.
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Ex 1. Classify each polynomial by degree and by
number of terms. First, combine Like terms with
the same letter and exponent power. Second, what
is the highest exponent and name it by the
degree. Third, count how many terms there are and
name it.

• 2x3 - 3x 4x3 b. 2x3 3x4
  2x3 5
• c. x2 4 - 8x - 2x3 d.
  3x3 2 x3 6x5
• e. 5x 2x3 4x2 f. x5 4x3 - x5
  3x2 4x3

6x3 3x
Cubic Binomial
3x4 5
Quartic Binomial
2x3 6x5 2
Quintic Trinomial
Cubic Polynomial
Cubic Trinomial
3x2
Quadratic Monomial
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Adding and Subtracting Polynomials

• To add and subtract polynomials, combine like
  terms.
• The standard form of a polynomial expression is
  written with the exponents in descending order of
  degree.

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Ex 2. Find the sum and write it from highest to
lowest power.

• a. (-2x2 3x3 5x 4) (-2x3 7x 6)
• -5x3 2x2 12x 2
• b. (2x4 4x3 5x - 2) (-2x4 7x2 8x 10)
• 4x3 7x2 13x 12
• c. (6x3 3x2 4) (2x3 5x2 3x 10)
• 8x3 2x2 3x 14

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Ex 3. Find the difference and write it from
highest to lowest power.

• a. (-6x3 6x2 7x 1) ( 3x3 5x2 2x 8)
  
• -9x3 x2 9x 9
• b. ( 3x3 12x2 5x 1) (-x2 5x 8)
• 3x3 11x2 10x 7
• c. ( 5x2 6x 11) (-8x3 x2 2)
• 8x3 4x2 6x 13

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Graphing Polynomial FunctionsA polynomial
function is a function that is defined by a
polynomial expression.

• Ex 4. Graph each function. Describe its general
  shape.
• P(x) 3x3 - 5x2 - 2x 1
• Cubic S shape
• Q(x) x4 - 8x2
• Quartic W Shape
• c. P(x) -3x3 - 2x2 2x - 1
• Cubic S Shape

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Examine the shapes of the linear, quadratic,
cubic, and quartic functions shown below.

• Linear Quadratic Cubic
  Quartic
• Line U S
  W

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Writing Activities

• 1). Describe 2 different ways to classify
  polynomials. Include examples.
• 2). What are the degree and the leading
  coefficient of
  Explain.
• 3). Which of the following are like terms?
  
• Explain.

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Review Of Introduction To Polynomials