Exponents and Polynomials

1
Exponents and Polynomials
Chapter 12
2
Chapter Sections
12.1 Exponents 12.2 Negative Exponents and
Scientific Notation 12.3 Introduction to
Polynomials 12.4 Adding and Subtracting
Polynomials 12.5 Multiplying Polynomials 12.6
Special Products 12.7 Dividing Polynomials
3
12.1

• Exponents

4
Exponents

• Exponents that are natural numbers are shorthand
  notation for repeating factors.
• 34 3 3 3 3
• 3 is the base
• 4 is the exponent (also called power)
• Note by the order of operations that exponents
  are calculated before other operations.

5
Evaluating Exponential Expressions
Example

• Evaluate each of the following expressions.

34
3 3 3 3
81
(5)2
( 5)(5)
25
62
(6)(6)
36
(2 4)3
(2 4)(2 4)(2 4)
8 8 8
512
3 42
3 4 4
48
6
Evaluating Exponential Expressions
Example

• Evaluate each of the following expressions.

a.) Find 3x2 when x 5.
3x2 3(5)2
3(5 5)
3 25
75
b.) Find 2x2 when x 1.
2x2 2(1)2
2(1)(1)
2(1)
2
7
The Product Rule

• Product Rule (applies to common bases only)
• am an amn

Example
Simplify each of the following expressions.
32 34
324
36
3 3 3 3 3 3
729
x4 x5
x45
x9
z3 z2 z5
z325
z10
(3y2)( 4y4)
3 y2 ( 4) y4
12y6
3( 4)(y2 y4)
8
The Power Rule

• Power Rule
• (am)n amn

Example
Simplify each of the following expressions.
(23)3
29
512
233
(x4)2
x8
x42
9
The Power of a Product Rule

• Power of a Product Rule
• (ab)n an bn

Example
Simplify (5x2y)3.
53 (x2)3 y3
125x6 y3
(5x2y)3
10
The Power of a Quotient Rule

• Power of a Quotient Rule

Example
Simplify
53 (x2)3 y3
125x6 y3
(5x2y)3
11
The Power of a Quotient Rule

• Power of a Quotient Rule

Example
Simplify the following expression.
12
The Quotient Rule

• Quotient Rule (applies to common bases only)

Example
Simplify the following expression.
13
Zero Exponent

• Zero exponent
• a0 1, a ? 0
• Note 00 is undefined.

Example
Simplify each of the following expressions. 50
1
(xyz3)0
x0 y0 (z3)0
1 1 1 1
x0
(x0)
1
14
12.2

• Negative Exponents and Scientific Notation

15
Negative Exponents

• Using the quotient rule from section 3.1,

But what does x -2 mean?
16
Negative Exponents

• So, in order to extend the quotient rule to cases
  where the difference of the exponents would give
  us a negative number we define negative exponents
  as follows.
• If a ? 0, and n is an integer, then

17
Simplifying Expressions
Example

• Simplify by writing each of the following
  expressions with positive exponents or
  calculating.

Remember that without parentheses, x is the base
for the exponent 4, not 2x
18
Simplifying Expressions
Example

• Simplify by writing each of the following
  expressions with positive exponents or
  calculating.

Notice the difference in results when the
parentheses are included around ?3.
19
Simplifying Expressions
Example

• Simplify by writing each of the following
  expressions with positive exponents.

(Note that to convert a power with a negative
exponent to one with a positive exponent, you
simply switch the power from a numerator to a
denominator, or vice versa, and switch the
exponent to its positive value.)
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Summary of Exponent Rules

• If m and n are integers and a and b are real
  numbers, then

Product Rule for exponents am an amn
Power Rule for exponents (am)n amn
Power of a Product (ab)n an bn
Zero exponent a0 1, a ? 0
21
Simplifying Expressions
Simplify by writing the following expression with
positive exponents or calculating.
22
Scientific Notation

• In many fields of science we encounter very large
  or very small numbers. Scientific notation is a
  convenient shorthand for expressing these types
  of numbers.
• A positive number is written in scientific
  notation if it is written as a product of a
  number a, where 1 ? a lt 10, and an integer power
  r of 10.
• a ? 10r

23
Scientific Notation

• To Write a Number in Scientific Notation
• Move the decimal point in the original number to
  the left or right, so that the new number has a
  value between 1 and 10.
• Count the number of decimal places the decimal
  point is moved in Step 1.
• If the original number is 10 or greater, the
  count is positive.
• If the original number is less than 1, the count
  is negative.
• Multiply the new number in Step 1 by 10 raised to
  an exponent equal to the count found in Step 2.

24
Scientific Notation
Example
Write each of the following in scientific
notation.
Since we moved the decimal 3 places, and the
original number was gt 10, our count is positive 3.
4700 4.7 ? 103
Since we moved the decimal 4 places, and the
original number was lt 1, our count is negative 4.
0.00047 4.7 ? 10-4
25
Scientific Notation

• To Write a Scientific Notation Number in
• Standard Form
• Move the decimal point the same number of spaces
  as the exponent on 10.
• If the exponent is positive, move the decimal
  point to the right.
• If the exponent is negative, move the decimal
  point to the left.

26
Scientific Notation
Example
Write each of the following in standard notation.
Since the exponent is a positive 3, we move the
decimal 3 places to the right.
5.2738 ? 103
5273.8
Since the exponent is a negative 5, we move the
decimal 5 places to the left.
00006.45 ? 10-5
0.0000645
27
Operations with Scientific Notation
Multiplying and dividing with numbers written in
scientific notation involves using properties of
exponents.
Example
Perform the following operations.
(7.3 8.1) ? (10-2 105)
59.13 ? 103
59,130
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12.3

• Introduction to Polynomials

29
Polynomial Vocabulary

• Term a number or a product of a number and
  variables raised to powers
• Coefficient numerical factor of a term
• Constant term which is only a number
• Polynomial is a sum of terms involving variables
  raised to a whole number exponent, with no
  variables appearing in any denominator.

30
Polynomial Vocabulary

• In the polynomial 7x5 x2y2 4xy 7
• There are 4 terms 7x5, x2y2, -4xy and 7.
• The coefficient of term 7x5 is 7,
• of term x2y2 is 1,
• of term 4xy is 4 and
• of term 7 is 7.
• 7 is a constant term.

31
Types of Polynomials

• Monomial is a polynomial with 1 term.
• Binomial is a polynomial with 2 terms.
• Trinomial is a polynomial with 3 terms.

32
Degrees

• Degree of a term
• To find the degree, take the sum of the exponents
  on the variables contained in the term.
• Degree of a constant is 0.
• Degree of the term 5a4b3c is 8 (remember that c
  can be written as c1).
• Degree of a polynomial
• To find the degree, take the largest degree of
  any term of the polynomial.
• Degree of 9x3 4x2 7 is 3.

33
Evaluating Polynomials

• Evaluating a polynomial for a particular value
  involves replacing the value for the variable(s)
  involved.

Example
Find the value of 2x3 3x 4 when x ?2.
2(? 2)3 3(? 2) 4
2x3 3x 4
2(? 8) 6 4
? 6
34
Combining Like Terms

• Like terms are terms that contain exactly the
  same variables raised to exactly the same powers.

Warning!
Only like terms can be combined through addition
and subtraction.
Example

• Combine like terms to simplify.
• x2y xy y 10x2y 2y xy

11x2y 2xy 3y
(1 10)x2y (1 1)xy ( 1 2)y
35
12.4

• Adding and Subtracting Polynomials

36
Adding and Subtracting Polynomials

• Adding Polynomials
• Combine all the like terms.
• Subtracting Polynomials
• Change the signs of the terms of the polynomial
  being subtracted, and then combine all the like
  terms.

37
Adding and Subtracting Polynomials
Example
Add or subtract each of the following, as
indicated. 1) (3x 8) (4x2 3x 3)
3x 8 4x2 3x 3
4x2 3x 3x 8 3
4x2 5
2) 4 ( y 4)
y 4 4
y 8
4 y 4
3) ( a2 1) (a2 3) (5a2 6a 7)
a2 1 a2 3 5a2 6a 7

• 3a2 6a 11

a2 a2 5a2 6a 1 3 7
38
Adding and Subtracting Polynomials

• In the previous examples, after discarding the
  parentheses, we would rearrange the terms so that
  like terms were next to each other in the
  expression.
• You can also use a vertical format in arranging
  your problem, so that like terms are aligned with
  each other vertically.

39
12.5

• Multiplying Polynomials

40
Multiplying Polynomials

• Multiplying polynomials
• If all of the polynomials are monomials, use the
  associative and commutative properties.
• If any of the polynomials are not monomials, use
  the distributive property before the associative
  and commutative properties. Then combine like
  terms.

41
Multiplying Polynomials
Example
Multiply each of the following. 1) (3x2)( 2x)
(3)( 2)(x2 x)
6x3
2) (4x2)(3x2 2x 5)
3) (2x 4)(7x 5)
2x(7x 5) 4(7x 5)
14x2 10x 28x 20
14x2 18x 20
42
Multiplying Polynomials
Example

• Multiply (3x 4)2
• Remember that a2 a a, so (3x 4)2 (3x
  4)(3x 4).

(3x 4)2 (3x 4)(3x 4)
(3x)(3x 4) 4(3x 4)
9x2 12x 12x 16
9x2 24x 16
43
Multiplying Polynomials
Example
Multiply (a 2)(a3 3a2 7).
(a 2)(a3 3a2 7)
a(a3 3a2 7) 2(a3 3a2 7)
a4 3a3 7a 2a3 6a2 14
a4 a3 6a2 7a 14
44
Multiplying Polynomials
Example
Multiply (3x 7y)(7x 2y)
(3x 7y)(7x 2y)
(3x)(7x 2y) 7y(7x 2y)
21x2 6xy 49xy 14y2
21x2 43xy 14y2
45
Multiplying Polynomials
Example
Multiply (5x 2z)2
(5x 2z)2 (5x 2z)(5x 2z)
(5x)(5x 2z) 2z(5x 2z)
25x2 10xz 10xz 4z2
25x2 20xz 4z2
46
Multiplying Polynomials
Example
Multiply (2x2 x 1)(x2 3x 4)
(2x2 x 1)(x2 3x 4)
(2x2)(x2 3x 4) x(x2 3x 4) 1(x2 3x
4)
2x4 6x3 8x2 x3 3x2 4x x2
3x 4
2x4 7x3 10x2 x 4
47
Multiplying Polynomials

• You can also use a vertical format in arranging
  the polynomials to be multiplied.
• In this case, as each term of one polynomial is
  multiplied by a term of the other polynomial, the
  partial products are aligned so that like terms
  are together.
• This can make it easier to find and combine like
  terms.

48
12.6

• Special Products

49
The FOIL Method

• When multiplying 2 binomials, the distributive
  property can be easily remembered as the FOIL
  method.
• F product of First terms
• O product of Outside terms
• I product of Inside terms
• L product of Last terms

50
Using the FOIL Method
Example
Multiply (y 12)(y 4)
(y 12)(y 4)
Product of First terms is y2
(y 12)(y 4)
Product of Outside terms is 4y
(y 12)(y 4)
Product of Inside terms is -12y
(y 12)(y 4)
Product of Last terms is -48
F O I L
(y 12)(y 4) y2 4y 12y 48

• y2 8y 48

51
Using the FOIL Method
Example
Multiply (2x 4)(7x 5)
(2x 4)(7x 5)
14x2 10x 28x 20
14x2 18x 20
We multiplied these same two binomials together
in the previous section, using a different
technique, but arrived at the same product.
52
Special Products

• In the process of using the FOIL method on
  products of certain types of binomials, we see
  specific patterns that lead to special products.
• Squaring a Binomial
• (a b)2 a2 2ab b2
• (a b)2 a2 2ab b2
• Multiplying the Sum and Difference of Two Terms
• (a b)(a b) a2 b2

53
Special Products

• Although you will arrive at the same results for
  the special products by using the techniques of
  this section or last section, memorizing these
  products can save you some time in multiplying
  polynomials.

54
12.7

• Dividing Polynomials

55
Dividing Polynomials

• Dividing a polynomial by a monomial
• Divide each term of the polynomial separately by
  the monomial.

Example
56
Dividing Polynomials

• Dividing a polynomial by a polynomial other than
  a monomial uses a long division technique that
  is similar to the process known as long division
  in dividing two numbers, which is reviewed on the
  next slide.

57
Dividing Polynomials
Divide 43 into 72.
Multiply 1 times 43.
Subtract 43 from 72.
Bring down 5.
Divide 43 into 295.
Multiply 6 times 43.
Subtract 258 from 295.
Bring down 6.
Divide 43 into 376.
Multiply 8 times 43.
Subtract 344 from 376.
Nothing to bring down.
58
Dividing Polynomials
As you can see from the previous example, there
is a pattern in the long division
technique. Divide Multiply Subtract Bring
down Then repeat these steps until you cant
bring down or divide any longer. We will
incorporate this same repeated technique with
dividing polynomials.
59
Dividing Polynomials
Divide 7x into 28x2.
Multiply 4x times 7x3.
Subtract 28x2 12x from 28x2 23x.
Bring down 15.
Divide 7x into 35x.
Multiply 5 times 7x3.
Subtract 35x15 from 35x15.
Nothing to bring down.
So our answer is 4x 5.
60
Dividing Polynomials
Divide 2x into 4x2.
Multiply 2x times 2x7.
Subtract 4x2 14x from 4x2 6x.
Bring down 8.
Divide 2x into 20x.
Multiply -10 times 2x7.
Subtract 20x70 from 20x8.
78
Nothing to bring down.