Intermediate Algebra chapter 5 part 2

Addition, Subtraction, and Multiplication of Polynomials
Chapter 5

Polynomials

General form of a polynomial in x:

a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 + . . . + a₂x² + a₁x¹ + a₀,   where

• the a_i, 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
Name	Example	Degree	Note
Monomial	3x2	2	One term (mono)
Binomial	2x + 1	1	Two terms (bi)
Trinomial	x3 + 2x2 + x	3	Three terms (tri)
Polynomial	6x4 + 5x3 + 4x2 + x + 7	4	Many terms (poly)

Polynomials can be in more than one variable....

Examples of Polynomials in x and y
Name	Example	Degree	Note
Monomial	3x2y3	5	One term (mono)
Binomial	2xy + y2	2	Two terms (bi)
Trinomial	x3y4 + 2x3y + xy2	7	Three terms (tri)
Polynomial	6x4y + 5x3y2 + 4x2y3 + xy4 + 7y5	5	Many terms (poly)

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Adding Polynomials

Combine like terms.

Example:   add 3x2 + 2x + 1 and 5x2 - 7x
(3x2 + 2x + 1) + (5x2 - 7x)  =
=  3x2 + 2x + 1 + 5x2 - 7x	remove parentheses
=  8x2 - 5x + 1	add like terms

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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 - 1	remove parentheses
=  2x2 + 5x - 1	add like terms

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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 + 15	simplifying
=  8x2 + 22x + 15	combine 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 - 18	simplifying
=  8x3 - 22x2 + 27x - 18	combine like terms

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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.

Example	Note
(x + y)2 = x2 + 2xy + y2	x 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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