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Sunday, 24 March 2013

Intermediate Algebra chapter 5 part 2

Addition, Subtraction, and Multiplication of Polynomials
Chapter 5

Polynomials
General form of a polynomial in x:
anxn + an-1xn-1 + an-2xn-2 + . . . + a2x2 + a1x1 + a0,   where
  • the ai, i = 1, 2, . . ., n are real numbers
  • n is a whole number
Degree of a term is the sum of the exponents on the variables in the term.
The term 4x3y5 has degree 8 since 3 + 5 = 8.
Degree of a polynomial is the degree of the highest degree term.
To write a polynomial in descending order for a certain variable means to write the polynomial from the term with the highest exponent (in the certain variable) on the left descending to the term with the lowest exponent (in the certain variable) on the right.
Descending order in x:   3x2 - 2x + 1.
Examples of Polynomials in x
NameExampleDegreeNote
Monomial3x22One term (mono)
Binomial2x + 11Two terms (bi)
Trinomialx3 + 2x2 + x3Three terms (tri)
Polynomial6x4 + 5x3 + 4x2 + x + 74Many terms (poly)
Polynomials can be in more than one variable....
Examples of Polynomials in x and y
NameExampleDegreeNote
Monomial3x2y35One term (mono)
Binomial2xy + y22Two terms (bi)
Trinomialx3y4 + 2x3y + xy27Three terms (tri)
Polynomial6x4y + 5x3y2 + 4x2y3 + xy4 + 7y55Many terms (poly)

Adding Polynomials
Combine like terms.
Example:   add 3x2 + 2x + 1 and 5x2 - 7x
(3x2 + 2x + 1) + (5x2 - 7x)  =  
=  3x2 + 2x + 1 + 5x2 - 7xremove parentheses
=  8x2 - 5x + 1add like terms

Subtracting Polynomials
1)  Remove parentheses (distribute "-" through).
2)  Combine like terms.
Example:   subtract 3x2 + 2x + 1 from 5x2 - 7x
(5x2 - 7x) - (3x2 + 2x + 1) =  
=  5x2 + 7x - 3x2 - 2x - 1remove parentheses
=  2x2 + 5x - 1add like terms

Multiplying Polynomials
1)  Use distributive property to remove parentheses and multiply out.
  • FOIL only works when multiplying binomials--the distributive property works when multiplying any polynomials together.
2)  Combine like terms.
Example:   (2x + 3)(4x + 5)
(2x + 3)(4x + 5) =
=  2x(4x + 5) + 3(4x + 5)distributive property
=  (2x)(4x) + (2x)(5) + 3(4x) + 3(5)distributive property again
=  8x2 + 10x + 12x + 15simplifying
=  8x2 + 22x + 15combine like terms
Example:   (2x - 3)(4x2 - 5x + 6)
(2x - 3)(4x2 - 5x + 6) =
=  2x(4x2 - 5x + 6) - 3(4x2 - 5x + 6)distributive property
=  (2x)(4x2) - (2x)(5x) + (2x)(6) - 3(4x2) + 3(5x) - 3(6)distributive property again
=  8x3 - 10x2 + 12x - 12x2 + 15x - 18simplifying
=  8x3 - 22x2 + 27x - 18combine like terms

Special Forms
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
(a + b)(a - b) = a2 - b2
Note:  the a and b may be any algebraic expression.
ExampleNote
(x + y)2 = x2 + 2xy + y2x is a
y is b
(2x + 5)2 =
= (2x)2 + 2(2x)(5) + 52
= 4x2 + 20x + 25
2x is a
5 is b
(2x - 5)2 =
= (2x)2 - 2(2x)(5) + 52
= 4x2 - 20x + 25
2x is a
5 is b
(3x2 - 2y)2 =
= (3x2)2 - 2(3x2)(2y) + (2y)2
= 9x4 - 12x2y + 4y2
3x2 is a
2y is b
(3x + 8)(3x - 8) =
= (3x)2 - 82 =
= 9x2 - 64
3x is a
8 is b
(4x2 - 3y)(4x2 + 3y) =
= (4x2)2 - (3y)2
= 16x4 - 9y2
4x2 is a
3y is b
25y2 - 81x4 =
= (5y)2 - (9x2)2
= (5y + 9x2)(5y - 9x2)
5y is a
9x2 is b

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