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Exponents
and
Polynomials
The Rules of Exponents
The Product Rule
The Product Rule
To multiply two exponential expressions that have the same base,
keep the base and add the exponents.
xa · xb = xa + b
Example:
Multiply.
a.) c4 · c5 = c4 + 5 = c9
b.) 3a3 · a6
c.) 4w2 · 2w5
Numerical
coefficient
= 3a3 + 6 = 3a9
= (4)(2)w2 + 5 = 8w7
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The Quotient Rule
The Quotient Rule (Part 1)
Use this form if the larger exponent is
a
a

b
x x
in the numerator and x  0.
b
x
Example:
Divide.
6
a.)
3  362  34
2
3
b.)
y 7  y 74  y 3
y4
Remember that the
base does not change.
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The Quotient Rule
The Quotient Rule (Part 2)
a
Use this form if the larger exponent is
x  1
b
ba
in the denominator and x  0.
x
x
Example:
Divide.
2
a.)
3  1  1
39 392 37
b.)
z  1   1
z11  z116
z5
Remember that the
base does not change.
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5
The Quotient Rule
The Quotient Rule (Part 3)
xa  x0  1
xa
if x  0 (00 remains undefined).
Example:
Divide.
a.)
1512  150  1
12
15
b.)
a 3b0  a 32 (1)  a
2
a
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Raising Exponential Expressions to a Power
Raising a Power to a Power
To raise a power to a power, keep the same base and multiply the
exponents.
 
x
a b
 x ab
Example:
Simplify.
a.) (x5)3 = x5·3 = x15
b.) (y3)3 = y3·3 = y9
7
Product Raised to a Power
Product Raised to a Power
a a
xy

x
y
 
a
Example:
Simplify.
a.) (2c)3 = (2)3c3 = 8c3
b.) (5xy)2
c.) (4x3y2)3
= (5)2(xy)2 = 25x2y2
= 43x9y6 = 64x9y6
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Quotient Raised to a Power
Quotient Raised to a Power



a
x   xa
y 
ya
if y  0.
Example:
Simplify.
4
a.)



x   x4
y  y 4
b.)
 3a   3  a   81a12
 2 
8
2 4
b
 b 
b 
3 4
4
3 4
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Negative Exponents &
Scientific Notation
Negative Exponents
Definition of a Negative Exponent
x  n  1n
x
if x  0.
Example:
Write with positive exponents.
4
 14
h
a.)
h
b.)
 
2a
3 5

1
 2a 
3 5

1
15
32a
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Negative Exponents
Properties of Negative Exponents
1  xn
x n
m
x
y n
yn
 m
x
Example:
Simplify. Write the expression with no negative exponents.
3 5
a.)
3 2
5
x y
x
x
x

 6
2
5
x y
yy
y
b.)


4 2 3
2ab c
 2 3 a 3b( 4)( 3) c (2)( 3)
12
b
3 3 12 6 
2 a b c
2 3 a 3c 6
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Scientific Notation
Scientific Notation
A number is written in scientific notation if it is in the form a × 10n,
where 1  a  10 and n is an integer.
8200 =
8.2
 1000 = 8.2  103
Greater than 1
and less than 10
Power
of 10
34,200,000 = 3.42  10000000 = 3.42  107
Scientific notation
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Decimal Notation to Scientific Notation
Example: Write 67,300 in scientific notation.
What power?
67,300. = 6.73  10n
Starting position
of decimal point
Ending position
of decimal point
The decimal point was moved 4 places to the left, so
we use a power of 4.
67,300 = 6.73  104
A number that is larger than 10 and written in scientific notation will
always have a positive exponent as the power of 10.
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Decimal Notation to Scientific Notation
Example: Write 0.048 in scientific notation.
What power?
0.048 = 4.8  10n
Starting position
of decimal point
Ending position
of decimal point
The decimal point was moved 2 places to the right, so
we use a power of –2.
0.048 = 4.8  10–2
A number that is smaller than 1 and written in scientific notation
will always have a negative exponent as the power of 10.
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Scientific Notation to Decimal Notation
Example: Write 9.1  104 in decimal notation.
9.1  104 = 9.1000  104 = 91,000
Move the decimal point
4 places to the right.
Example: Write 6.72  10–3 in decimal notation.
6.72  10–3 =
6.72  10–3 = 0.00672
Move the decimal point
3 places to the left.
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Fundamental Polynomial
Operations
Recognizing Polynomials
A polynomial in x is the sum of a finite number of terms of the
form axn , where a is any real number and n is a whole number.
5x2 + x – 3,
1.2c3 + 5.1
A multivariable polynomial is a polynomial with more than one
variable.
4y3z5 + 7w2 + 6, 2a3 + b2 – ab
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Degrees of Terms & Polynomials
The degree of a term is the sum of the exponents of all of the
variables in the term.
5x2
The degree is 2.
The degree is 3.
2c3 + 5.1
The degree of a polynomial is the highest degree of all of the terms
in the polynomial.
The degree is 8.
4y3z5 + 7w2 + 6
The degree is 3.
2a3 + b2 – ab
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Types of Polynomials
A monomial has one term.
5x2, a, 2c5d
A binomial has two terms.
5x2 + 3, ab – 4 , 2 + c
A trinomial has three terms.
5x2 + x – 3, ab – a + b , –c2 + c + 2
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Adding Polynomials
To add polynomials, add the like terms.
Example:
Add. (5x2 – x + 6) + (2x2 – 3)
(5x2 – x + 6) + (2x2 – 3)
= [5x2 + 2x2] + (– x ) + [6 + (– 3)]
= 7x2 + (– x ) + 3
= 7x2 – x + 3
A polynomial is written in decreasing order when each exponent
is decreasing.
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Subtracting Polynomials
To subtract two polynomials, change the sign of each term in the
second polynomial and then add the like terms.
Example:
Subtract. (3a4 + 5a – 6) – (2a4 + 2a – 3)
(3a4 + 5a – 6) – (2a4 + 2a – 3)
= (3a4 + 5a – 6) + (– 2a4 – 2a + 3)
= (3a4 – 2a4) + (5a – 2a) + (–6 + 3)
= a4 + 3a – 3
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Evaluating Polynomials
Polynomials may be used to predict a value. This is done by
evaluating the polynomial.
Example:
Evaluate the polynomial. Let a = 3.
– 2a4 + 2a – 3
– 2(3)4 + 2(3) – 3
= – 2(81) + 6 – 3
= – 162 + 6 – 3
= – 159
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Multiplying Polynomials
Multiplying Polynomials
Use the distributive property to multiply a monomial by a
polynomial.
Example:
a.) Multiply. 3(c – 4)
3(c – 4) = 3(c) + 3(– 4)
= 3c – 12
b.) Multiply. 2x3(x2 – x + 2)
2x3(x2 – x + 2) = 2x3(x2) + 2x3(– x) + 2x3(2)
= 2x5 – 2x4 + 4x3
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Multiplying Two Binomials
To multiply two binomials, the distributive property is used so that
every term in one polynomial is multiplied by every term in the
other polynomial.
Example: Multiply. (7x + 3)(2x + 4)
(7x + 3)(2x + 4) = (7x + 3)(2x) + (7x + 3)(4)
= 14x2 + 6x + 28x + 12
= 14x2 + 34x + 12
This method used to multiply two binomials is referred to as the
FOIL method.
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The FOIL Method
Consider (a + b)(c + d):
F
“First”
Multiply the first terms together.
F
(a + b) (c + d)
O
“Outer” Multiply the outer terms together.
O
(a + b) (c + d)
I
product ad
“Inner” Multiply the inner terms together.
I
(a + b) (c + d)
L
product ac
“Last”
product bc
Multiply the last terms together.
L
(a + b) (c + d)
product bd
The product of the two binomials is the sum of these four products:
(a + b)(c + d) = ac + ad + bc + bd
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The FOIL Method
Example: Multiply using the FOIL method.
(7x + 3)(2x + 4)
L
F
(7x + 3)(2x + 4)
I
O
F
O
I
L
= (7x)(2x) + (7x)(4) + (3)(2x) + (3)(4)
= 14x 2
+ 28x
+ 6x + 12
= 14x 2
+ 34x
+ 12
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The FOIL Method
Example: Multiply. (3x + 4)(5x  y)
(3x + 4)(5x  y)
= (3x)(5x) + (3x)( y) + (4)(5x) + (4)( y)
F
O
I
L
= 15x2 + ( 3xy) + 20x + ( 4y)
= 15x2  3xy + 20x  4y
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Multiplication: Special
Cases
Multiplying Binomials: (a + b)(a  b)
Example: Multiply. (2x + 4)(2x  4)
(2x + 4)(2x  4)
= (2x)(2x) + (2x)( 4) + (4)(2x) + (4)( 4)
F
O
I
L
= 4x2 + ( 8x) + 8x + ( 16)
The inner and outer products cancel.
= 4x2  16
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Multiplying Binomials: (a + b)(a  b)
Multiplying Binomials: A Sum and a Difference
(a + b)(a  b) = a2 – b2
Example: Multiply. (5a + 3)(5a  3)
(5a + 3)(5a  3)
= (5a)2  32
= 25a2  9
Example: Multiply. (8c + 2d)(8c  2d)
(8c + 2d)(8c  2d)
= (8c)2  (2d)2
= 64c2  4d2
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Multiplying Binomials: (a + b)2 and (a  b)2
Example: Multiply. (x + 6)2
(x + 6)2
= (x + 6)(x + 6)
= (x)(x) + (x)(6) + (6)(x) + (6)(6)
F
O
I
L
= x2 + 6x + 6x + 36
The inner and outer products are the same.
= x2 + 12x + 36
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Multiplying Binomials: (a + b)2 and (a  b)2
Squaring Binomials
(a + b)2 = a2 + 2ab + b2
(a  b)2 = a2 – 2ab + b2
Example: Multiply. (12a  3)2
(12a  3)2
= (12a)2  2(12a)(3) + (3)2
= 144a2  72a + 9
Example: Multiply. (x + y)2
(x + y)2 = x2 + 2xy + y2
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Multiplying Polynomials with More
than One Term
To multiply any two polynomials, vertical multiplication may be used.
Example: Multiply. (w – 1)(2w2 + 7w + 3)
2w2 + 7w + 3
w–1
Keep the terms
lined up.
Write vertically,
lining up the terms.
– 2w2 – 7w – 3
2w3 + 7w2 + 3w
Multiply – 1(2w2 + 7w + 3).
2w3 + 5w2 – 4w – 3
Add the terms in each column.
Multiply w(2w2 + 7w + 3).
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Multiplying Polynomials with More
than One Term
To multiply any two polynomials, the distributive property may also be
used.
Example: Multiply. (w – 1)(2w2 + 7w + 3)
(w –
1)(2w2
+ 7w + 3)
Multiply the first term in the first polynomial
by every term in the second polynomial…
= w(2w2) + w(7w) + w(3) + (– 1)2w2 + (– 1)7w +(– 1)3
… and multiply the second term in the first
polynomial by every term in the second polynomial.
= 2w3 + 7w2 + 3w +(– 2w2) + (– 7w) + (– 3)
= 2w3 + 5w2 – 4w – 3
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