DEV 108 Unit 3 Notes: Exponent Rules Polynomials

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DEV 108Unit 3 NotesExponent RulesPolynomials
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Exponent Review
Exponential Notation represents repeated
multiplication.
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Exponent
5
Translates to
Base
5 x 5 x 5 125
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Exponent Review
Multiplying with Exponential Values If there
ARE parenthesis Exponent is even positive
answer (-5)2 (-5)(-5) 25 Exponent is odd
negative answer (-5)3 (-5)(-5)(-5) -125
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Multiplying with Exponential Values
Multiplying with exponential valuesIf there are
NO parenthesis Exponent only applies to the
value it touches - 52 (-1)(5)(5) -25 -
53 (-1)(5)(5)(5) -125
(The parenthesis are used to represent
multiplication)
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Multiplying with Exponents
translates to
x3 x2
x x x
x x
x5
translates to
y2 y2
y y
y y
y4
When you multiply where the base values are the
same
Step 1 Write the base value (variable) in
your answer
Step 2 ADD the exponential values together and
write as the exponent in your answer
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Multiplying with Exponents Examples
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Power Rules with Exponents
2
(x3)
x3 x3
translates to
x6
2
(3x2y3)
(3x2y3)(3x2y3)
translates to
9x4y6
When you have a power raised to a power
Step 1 Write the base value (variable) in
your answer
Step 2 MULTIPLY the exponential values
together and write as the exponent in your
answer
Remember the exponent also applies to the
numerical coefficient
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Power Rules with Exponents Examples
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Division with Exponents
x x x x
x2
x4 x2
x x
a a
1 a3
a2 a5
a a a a a
a b c
1
abc abc
a b c
When dividing with exponents
Step 1 Divide numerical coefficients if necessary
Step 2 Subtract the exponential values
Be careful where you put your answers!
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Division with Exponents Examples
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Polynomials
Polynomials are mathematical classifications for
expressions
Expressions cannot be classified as a polynomial
if it contains - Negative exponent - Variable
in the denominator
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Classifications of Polynomials
Monomials Expressions with 1 term
Examples 8x -4y 15r
Binomials Expressions with 2 terms
Examples 2y 3 5x 2y -5x2 3x
Trinomials Expressions with 3 terms
Examples x2 - 3x 10 4x2 8x 12y
Polynomials Expressions with 3 or more terms
Examples 5x2 10x 5y 15y2 9a2 12a
15ab 6b2
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Adding Polynomials
To add polynomials, add like terms. Be sure to
remove parenthesis before combining like terms.
(3x2 4x) (5x2 8x) 8x2 12x (5x2 4x
3) (2x2 7x 6) 7x2 11x 3
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Subtracting Polynomials
When subtracting polynomials, add the opposite
values. In other words, you change the signs in
the second polynomial (the one being subtracted)
then combine like terms.
(6x 7) (2x 8) means 6x 7 2x 8
means 6x 2x 7 8 4x 1
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Subtracting Polynomials
(8x 4) (5x 8) means 8x 4 5x 8
means 8x 5x 4 8 3x - 12
The only time that the sign inside the
parenthesis changes is when you change from a
subtraction problem to an addition problem
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Adding/Subtracting Polynomial Practice
Practice Add or subtract each. (9x 3) (-7x
4) (9y 12) (12y 8) (15x - 3) (5x2
4x) (y2 y - 1) (4y2 6)
2x 1 21y 20 5x2 19x - 3
5y2 y - 7
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Adding/Subtracting Polynomial Practice
Practice Add or subtract each. (7a 9) (4a
6) (12x 5) (-x 14) (2x 12) (4x
6) (-7y 9) (6y 2)
3a - 15 13x 19 -2x 18 -13y 7
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Evaluating Polynomials
When evaluating polynomials, you will be given a
value for the variable. Plug the value into the
problem then chug through the arithmetic. Find
the value for the polynomial 4x2 5x 3 when
x 3 4x2 5x 3 (4)(3)(3) (5)(3)
3 36 - 15 3 24
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Evaluating Polynomials Practice
Solve. Let x 2, y -2, and a 5 3x
10 4y2 5y 6a 3.5 2x2 5xy 3y2 a2
10a 25 ½xy 2a
16 26 33.5 40 100 -12
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Evaluating Polynomials Practice
Solve. Let x 2, y -2, and a 5 0.75a x3
(y)2 a3 2a 3y 2.5a 10y
7.75 141 32.5
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Multiplying Binomials
Use the FOIL method to multiply binomials
F first values
O outside values
I inside values
L last values
6x
3
8x2
4x
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Multiplying Binomials
Once you have done all of your multiplying,
combine like termsif you follow the FOIL
method, the like terms will be the two values
in the middle.
8x2 10x 3
This is your final answer
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Multiplying Binomials
Another way to multiply binomials is to use a
more traditional multiplication set-up
Hint be sure to put your signs in to know
which values are positive and which are
negative
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Patterns for Multiplying Binomials
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Multiplying Binomials Practice
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Multiplying Binomials Practice
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Factoring Polynomials
When factoring polynomials, pay attention to what
you are being asked to factor. Step 1 The
greatest common factor or greatest common
monomial factor is factored out (divided out)
and written in front or a set of
parenthesis Step 2 The remaining values are
written inside the parenthesis
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Greatest Common Factor (GCF)
GCF Greatest Common Factor Factor out (divide
out) the largest common number 4x 10y
6x 9y 2(2x 5y) 3(2x 3y) 4a
20b 4(a 5b)
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Factoring out the GCF Practice
8x 14y 4a 8b 12r 18s 5x2 10x
15 21x2 14x 7 12x2 18x 24
2(4x 7y) 4(a 2b) 6(2r 3s) 5(x2 2x
3) 7(3x2 2x 1) 6(2x2 3x 4)
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Greatest Common Monomial Factor (GCMF)
GCMF Greatest Common Monomial Factor Factor out
(divide out) the largest common number and
variable 8a2 16ab 8a(a 2b) 12x3
18x2 6x 6x(2x2 3x 1)
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Factoring out the GCMF Practice
8a2 16ab 18x3 6x2 14x 9x3 6x2y
12xy 16x3 8x2 8x 18a2b 8a 5a2b2
10ab 15a2b
8a(a 2b) 2x(9x2 3x 7) 8x(2x2 x
1) 8x(2x2 x 1) 2a(9ab 4) 5ab(ab 2
3a)
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Review of Polynomials
Polynomials are expressions with 3 or more terms.
Factoring is the inverse of FOILing. Factoring
is working backwards to get two binomials.
Always write polynomials in descending order of
degrees.
x3 x2 x
There are many different ways to factor
polynomials.
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Factoring Polynomials ABC Method
The letters a, b, and c represent various
positions in an equation
Multiply a and c
1 2 2
Find factors for the product 2
_x2 3x 2
1 2
-1 -2
Identify the pair of factors, when added
together, give you the b value 3
These are your factors put them in parenthesis
with your variable
(x 1)(x 2)
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Factoring Polynomials ABC Method
_x2 4x 3
Multiply a and c
1 3 3
Find factors for the product 3
These are your factors put them in parenthesis
with your variable
1 3
-1 -3
Identify the pair of factors, when added
together, give you the b value 4
(x 1)(x 3)
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Factoring Polynomials ABC Method
Multiply a and c
_x2 5x 6
1 6 6
Find factors for the product 6
1 6
-1 -6
2 3
-2 -3
These are your factors put them in parenthesis
with your variable
Identify the pair of factors, when added
together, give you the b value 5
(x 2)(x 3)
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Factoring Polynomials ABC Method
Multiply a and c
_x2 _x - 2
1 -2 -2
Find factors for the product -2
1 -2
-1 2
These are your factors put them in parenthesis
with your variable
Identify the pair of factors, when added
together, give you the b value 1
(x - 1)(x 2)
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Factoring Polynomials ABC Method
Multiply a and c
_x2 - 4x 4
1 4 4
Find factors for the product 4
-1 -4
1 4
-2 -2
2 2
These are your factors put them in parenthesis
with your variable
Identify the pair of factors, when added
together, give you the b value -4
(x - 2)(x - 2)
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Factoring Polynomials ABC Method
Multiply a and c
_x2 - 4x 3
1 3 3
Find factors for the product 3
-1 -3
1 3
These are your factors put them in parenthesis
with your variable
Identify the pair of factors, when added
together, give you the b value -4
(x - 1)(x - 3)
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Factoring Polynomials ABC Method
Multiply a and c
_x2 4x 4
1 4 4
Find factors for the product 4
-1 -4
1 4
-2 -2
2 2
These are your factors put them in parenthesis
with your variable
Identify the pair of factors, when added
together, give you the b value 4
(x 2)(x 2)
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Factoring Polynomials ABC Method
Multiply a and c
_x2 9x 20
1 20 20
Find factors for the product 20
-1 -20
1 20
-2 -10
2 10
These are your factors put them in parenthesis
with your variable
-4 -5
4 5
Identify the pair of factors, when added
together, give you the b value 9
(x 4)(x 5)
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Factoring Polynomials ABC Method
Multiply a and c
_x2 4x - 12
1 -12 -12
Find factors for the product 20
1 -12
-1 12
2 -6
-2 6
These are your factors put them in parenthesis
with your variable
3 -4
-3 4
Identify the pair of factors, when added
together, give you the b value 4
(x - 2)(x 6)
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Review the things that you need to review.
Study the things that you need to spend more time
on.
Ask questions about things you dont understand.
PRACTICEPRACTICEPRACTICE