Download Appendix A.2 - Thomas Hauner

Appendix A
Basic Algebra
Review
Section A-2
Operations on
Polynomials
Operations on Polynomials
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Natural Number Exponents
Polynomials
Combining Like Terms
Addition and Subtraction
Multiplication
Combined Operations
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Natural Number Exponent
Natural Number Exponent
For n a natural number and b any real number,
bn  b b ...b
n factors of b
where n is called the exponent and b is called the base.
First Property of Exponents
For any natural numbers m and n, and any real number b,
b m bn  b mn
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Polynomials
Algebraic expressions
Constants and variables and the algebraic operations of
addition, subtraction, multiplication, division, raising to
powers, and taking roots. Special types of algebraic
expressions are called polynomials.
Polynomial in one variable x
Adding or subtracting constants and terms of the form axn,
where a is a real number and n is a natural number.
Polynomial in two variables x and y
Adding or subtracting constants and terms of the form axmyn,
where a is a real number and m and n are a natural numbers.
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Polynomials
Degree of the term
Power of the variable.
Degree of the term with two or more variables
Sum of the power of the variables.
Degree of the polynomial
Degree of the nonzero term with the highest degree in the
polynomial.
Polynomial of degree 0
A nonzero constant. The number 0 is also a polynomial but
is not assigned a degree.
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Combining Like Terms
A constant in a term of a polynomial, including the
sign that precedes it, is called the numerical
coefficient, or simply, the coefficient, of the term.
Distributive Properties of Real Numbers
1. a b  c   b  c a  ab  ac
2. a b  c   b  c a  ab  ac
3. a b  c  ...  f   ab  ac  ...  af
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Combining Like Terms
Two terms in a polynomial are called like terms if they have
exactly the same variable factors to the same powers.
If a polynomial contains two or more like terms, these terms
can be combined into a single term by making use of
distributive properties.
3x 2 y  5xy 2  x 2 y  2x 2 y  3x 2 y  x 2 y  2x 2 y  5xy 2


 3x 2 y  x 2 y  2x 2 y  5xy 2
 3  1 5 x 2 y  5xy 2
 2x 2 y  5xy 2
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Addition and Subtraction
Addition and subtraction of polynomials can be thought of in
terms of removing parentheses and combining like terms.
Add horizontally:
x 4  3x 3  x 2 ,  x 3  2x 2  3x, 3x 2  4x  5
x
4


 3x 3  x 2  x 3  2x 2  3x  3x 2  4x  5

 x 4  3x 3  x 2  x 3  2x 2  3x  3x 2  4x  5
 x 4  4x 3  2x 2  x  5
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Addition and Subtraction
We can also add vertically:
Add:
x 4  3x 3  x 2 ,  x 3  2x 2  3x, 3x 2  4x  5
x 4  3x 3  x 2
 x 3  2x 2  3x
3x 2  4x  5
x 4  4x 3  2x 2  x  5
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Multiplication
Multiplication of algebraic expressions involves the extensive
use of distributive properties for real numbers, as well as
other real number properties.
Multiply:
2x  33x 2  2x  3
2x  33x 2  2x  3 2x 3x 2  2x  3 33x 2  2x  3
 6x 3  4x 2  6x  9x 2  6x  9
 6x 3  13x 2  12x  9
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Multiplication
Thus, to multiply two polynomials, multiply each term of one
by each term of the other, and combine like terms.
For the product of two binomials we use this process:
2x  13x  2   6x 2  4x  3x  2
 6x 2  x  2
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Special Products
Special products:
1.
a  b a  b   a 2  b 2
2.
a  b   a 2  2ab  b 2
3.
a  b 2  a 2  2ab  b 2
2
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Combined Operations
Note that in simplifying, we usually remove grouping symbols
starting from the inside. That is, we remove parentheses ( ) first,
then brackets [ ], and finally braces { }, if present.
Order of Operations
Multiplication and division precede addition and subtraction,
and taking powers precedes multiplication and division.



3x  5 3  x  x 3  x   3x  5  3  x  3x  x 2 

 3x  5  3x  9x  3x 2


 3x  5  3x  9x  3x 2
 3x 2  3x  5
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