Polynomials

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Polynomials
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In This Unit

• Review simplifying polynomials, distributive
  property exponents
• Classifying Polynomials
• Area Perimeter with Algetiles
• Factoring in Algebra
• Multiplying Dividing Monomials
• Multiplying Polynomials Monomials
• Factoring Polynomials
• Dividing Polynomials
• Multiplying Two Binomials
• Factoring Trinomials

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• A monomial is a number, a variable, or a product
  of numbers and variables.
• A polynomial is a monomial or a sum of monomials.
  The
• exponents of the variables of a polynomial must
  be positive.
• A binomial is the sum of two monomials, and a
  trinomial is the sum of three monomials.
• The degree of a monomial is the sum of the
  exponents of its variables.
• To find the degree of a polynomial, you must find
  the degree of each term. The greatest degree of
  any term is the degree of the polynomial. The
  terms of a polynomial are usually arranged so
  that the powers of one variable are in ascending
  or descending order.

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Classifying Polynomials

• A monomial is an expression with a single term.
  It is a real number, a variable, or the product
  of real numbers and variables.
• Example 4, 3x2, and 15xy3 are all monomials

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Classifying Polynomials

• A binomial is an expression with two terms. It is
  a real number, a variable, or the product of real
  numbers and variables.
• Example 3x 9

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Classifying Polynomials

• A trinomial is an expression with three terms. It
  is a real number, a variable, or the product of
  real numbers and variables.
• Example x2 3x 9
• Now you try to Classify Each?

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POLYNOMIAL Monomial Binomial Trinomial
2x 9 x
3 x
10x2 2x 9 x
2(x 4) x
3x 4 x
6x - 8 x
-9x x
3x2 3xy 9x x
10 x
2x x
x2 3xy 9xyz x
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Algebra Tiles Area
x
x
1
x
1
1
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Draw algebra tiles to represent the polynomial
3x2 2x 5
Recall This Algebraic Expression has 3
Terms 3x2 3 is the coefficient, x2 is
the variable 2x -2 is the coefficient, x
is the variable 5 5 is the
constant term
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What is the area of a rectangle?
AREA Length x Width
How do you find the perimeter of a rectangle?
ADD up all of the sides
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We can combine algebra tiles to form a rectangle.
We can then write the area and the perimeter
of the rectangle as a polynomial.
This rectangle has the following properties
Length 5 Width x Perimeter is x 5
x 5 2x 10 Area LW 5 x x 5x
x
5
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Determine the Area Perimeter of the following
Rectangles
x
x
x
x
This rectangle has the following properties
Length 3x Width x Perimeter x
3x x 3x 8x Area (3x) (x) 3x2
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2.)
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2.)
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Multiplying MonomialsRECALL

• Multiplying Powers When multiplying powers with
  the same base we add the exponents
• Example x2 x2 x4
• Dividing Powers When dividing powers with the
  same base we subtract the exponents
• Example x3 x1 x2
• Power of a Power
• Example

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Multiplying Monomials

• (3x2)(5x3) (3 x x) (5 x x x)
• (3) (5) (xxxxx)
• 15x5

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With Algetiles
x x x2
(2)(5x) 10x
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Prime Factor Review

• A prime factor is a whole number with exactly TWO
  factors, itself and 1
• A composite number has more than two factors

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FACTOR TREES
So 3 x 2 x 2 are prime factors of 12
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Practice Exercises

• Express each number as a product of its prime
  factors
• 30
• 36
• 25
• 42
• 75
• 100
• 121
• 150

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Practice Solutions

• 2 x 3 x 5
• 2 x 2 x 3 x 3
• 5 x 5
• 2 x 3 x 7
• 3 x 5 x 5
• 2 x 2 x 5 x 5
• 11 x 11
• 2 x 3 x 5 x 5

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We can factor in algebra too?

• 3x2 3 x x
• 5x 5 x
• 2x4 2 x x x x
• 2x2y2 2 x x y y
• Lets Try
• a)4x3 b) x2 c)2x6
• d) 9x2y e) -6a2b2

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We can factor in algebra too?

• a)4x3 4 (x x x )
• b) x2 (-1) (x x)
• c)2x6 (2) (x x x x x x)
• d) 9x2y (9 2) (x) (y)
• e) -6a2b2 (-6) (a a) (b b)

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Greatest Common Factor

• The greatest of the factors of two or more
  numbers is called the greatest common factor
  (GCF).
• Two numbers whose GCF is 1 are relatively prime.

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Finding the GCF

• To find the GCF of 126 and 60.
• 2 x 3 x 3 x 7
• 60 2 x 2 x 3 x 5
• List the common prime factors in each list 2, 3.
  
• The GCF of 126 and 60 is 2 x 3 or 6.

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Finding the GCF
Find the GCF of 140y2 and 84y3 140y2 2 2
5 7 y y 84y3 2 2 3 7 y y
y List the common prime factors in each list 2,
2, 7, y, y The GCF is 2 2 7 y y 28y2

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Finding the GCF

• Try These Together
• What is the GCF of 14 and 20?
• 2. What is the GCF of 21x4 and 9x3?
• HINT Find the prime factorization of the numbers
  and then find the product of their common
  factors.

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Finding the GCF

• What is the GCF of 14 and 20?
• Factors of 14 2, 7
• Factors of 20 2, 4, 5, 10
• Therefore the GCF is 2
• 2. What is the GCF of 21x4 and 9x3?
• Factors of 21x4 3 7 x x x x
• Factors of 9x3 3 3 x x x
• Therefore the GCF is 3 x x x 3x3