Positive and Negative Integral
Exponents
Integral Exponents, Polynomials,
and Factoring
Peter Lo
M014 © Peter Lo 2002
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Product Rule for Exponents
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Quotient Rule for Exponent
If m and n are integers and a ≠ 0, then
u am·an = a m+n
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Example:
u 34 ·36
u -2y -3 (-5y -4 )
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If m and n are any integers and a ≠ 0, then
am
= a m− n
an
Example:
u
3
m5
m −3
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Zero Exponent
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If a is any non-zero real number, then
u a0 = 1
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Example:
u -2a 5 b 6 ·3a -5 b -2
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Scientific Notation
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Raising an Exponential Expression
to a Power
Raising a Product to a Power
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Example:
u (23 )5
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Example:
u (-3x)4
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Raising a Quotient to a Power
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Summary
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Example
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Polynomials
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A Polynomial is defined as single term or a sum
of a finite number of terms.
u P(x) = an xn + an-1 xn-1 + … + a2 x2 + a1 x + a0
Term is a single number of the product of a
number or one or more variables raised to whole
number powers.
The number preceding the variable in each term is
called the Coefficient.
A number is referred to as a Constant.
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Operation of Polynomials
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Example
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Example
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Find the product (x + 2) (x2 + 3x – 5)
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Multiply (3x2 + 4)(x2 – 7x + 2)(x + 3)
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Find the sums
u (x2 – 5x – 7) + (7x2 – 4x + 10)
u (3x3 – 5x2 – 7) + (4x2 – 2x + 3)
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Find the differences
u (x2 – 7x – 2) – (5x2 + 6x – 4)
u (6y 3 z – 5yz + 7) – (4y 2 z – 3yz – 9)
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Multiplying Binomials (FOIL)
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Multiplying Binomials (FOIL)
Square of Binomial
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Rule for the Square of a Sum
u (a + b)2 = a2 + 2ab + b 2
Rule for the Square of a Difference
u (a – b)2 = a2 – 2ab + b 2
Rules of a Sum and a Difference
u (a + b)(a – b) = a2 – b 2
M014 © Peter Lo 2002
Example
Division of Polynomials
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Multiply (3x + 2y)(3x – 2y)
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Multiply (3x + 2y)(3x – 2y)
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Divide 4x3 – x – 9 by 2x – 3.
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Synthetic Division
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Example
When dividing a polynomial by a binomial of
form x – c, we can use Synthetic Division to speed
up the process. For Synthetic Division, we write
only the essential parts of ordinary division.
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Strategy for using Synthetic
Division
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Factoring Polynomials
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Divide x3 – 5x2 + 4x – 3 by x – 2.
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Factoring out the Great Common Factor
Factoring out the Opposite of the Great Common
Factor
Factoring the Difference of Two Squares
Factoring Perfect Square Trinomial
Factoring a Different or a Sum of Two Cubes
Factoring a Polynomial Completely
Factoring by Substitution
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Factoring out the Great Common
Factor
Factoring Perfect Square
Trinomials
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Example:
u Factorize 18x3 – 6x2 .
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Factorize 9y 2 – 64.
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Factoring a Difference or a Sum of
Two Cube
Example
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The trinomial that results from squaring a
binomials is call Perfect Square Trinomial.
u a2 + 2ab + b 2 = (a + b)2
u a2 – 2ab + b 2 = (a – b)2
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a3 – b 3 = (a – b)(a 2 + ab + b 2 )
a3 + b 3 = (a + b)(a 2 – ab + b 2 )
Example:
u Factorize 27x3 + 64
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Factoring ax2 + bx + c
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Factoring Strategy
Strategy for factoring ax2 + b x + c by the ac
Method:
u To factor the trinomial ax2 + b x + c
t Find two integers that have a product equal
to ac and a sum equal to b.
t Replace bx by two terms using the two new
integers as coefficients.
t Then factor the resulting four-term
polynomial by grouping
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Example
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The Factor Theorem
Factorize 3x8 – 243 completely
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Reference
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Algebra for College Students (Ch. 5)
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