Matho083 Bianco

1
Matho083 Bianco Warm Up
Multiply 1) (x2) (x3) 2) (3x2) (4x3) 3) (4x5)
(-2x3) 4) (5xy2) (2x3y)
2
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

Scientific notation
3
Scientific Notation

• Writing a Number in Scientific Notation
• Move the decimal point in the original number to
  the until the new number has a value between 1
  and 10.
• Count the number of decimal places the decimal
  point was 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.
• Write the product of the new number in Step 1 by
  10 raised to an exponent equal to the count found
  in Step 2.

4
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
5
Scientific Notation

• Writing a Scientific Notation Number in Standard
  Form
• Move the decimal point the same number of places
  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.

6
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
7
SCIENTIFIC TO STANDARD

• Exp positive move decimal right
• Exp negative move decimal left

8
STANDARD TO SCIENTIFIC

• Move decimal right negative exp
• Move decimal left positive exp

9
JOURNAL

• Simplify
• (7x5)(x3)
• (32)43
• (p2q8)(p2q)
• (3pq6)2

10
PRODUCT OF SCIENTIFIC NOTATION

• Multiply the numbers
• Add the exponents

11
EXAMPLE

• (4.11 x 1013) x
• (3.78 x 10-5)

12
EXAMPLE 2

• (9 x 104) x
• (4 x 106)

13
QUOTIENT OF SCIENTIFIC NOTATION

• Divide the numbers
• Subtract the exponents

14
EXAMPLE

• (4.11 x 1013)
• (3.78 x 10-5)

15
EXAMPLE 2

• (9 x 104)
• (4 x 106)

16

• 1. Evaluate if x 3 and y -5
• x3 y4
• 2. Multiply
• x5y2 - x2y5
• 3. Mult 3x3 (-2 x5 )

17
5.3

• Polynomials and Polynomial Functions

18
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.

19
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.

20
Types of Polynomials

• Monomial is a polynomial with one term.
• Binomial is a polynomial with two terms.
• Trinomial is a polynomial with three terms.

21
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.

22
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
23
Adding Polynomials

• Adding Polynomials
• To add polynomials, combine all the like terms.

Example
Add. (3x 8) (4x2 3x 3)
3x 8 4x2 3x 3
4x2 3x 3x 8 3
4x2 5
24
Subtracting Polynomials

• Subtracting Polynomials
• To subtract polynomials, add its opposite.

Example
Subtract.
4 ( y 4)
y 4 4
y 8
4 y 4
( a2 1) (a2 3) (5a2 6a 7)
a2 1 a2 3 5a2 6a 7
3a2 6a 11
a2 a2 5a2 6a 1 3 7
25
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.

26
Types of Polynomials
Using the degree of a polynomial, we can
determine what the general shape of the function
will be, before we ever graph the function. A
polynomial function of degree 1 is a linear
function. We have examined the graphs of linear
functions in great detail previously in this
course and prior courses. A polynomial function
of degree 2 is a quadratic function. In general,
for the quadratic equation of the form y ax2
bx c, the graph is a parabola opening up when a
gt 0, and opening down when a lt 0.
27
Types of Polynomials
Polynomial functions of degree 3 are cubic
functions. Cubic functions have four different
forms, depending on the coefficient of the x3
term.
28
POLYNOMIALS
MULTIPLY 1. (7) (2x - 5) (7) (2x) - (7) (5)
14x - 35
29
POLYNOMIALS
MULTIPLY 2. (2x2)(5x4 7x) (2x2) (5x4) (2x2)
(7x) 10x6 14x3
30
Multiplying by FOIL
(3x 2) (5x 4) F O I L
31
Multiplying by FOIL
(3x 2) (5x 4) F O I L
32
Multiplying by FOIL
(3x 2) (5x 4) F O I L
33
Multiplying by FOIL
(3x 2) (5x 4) F O I L
34
Multiplying by FOIL
(3x 2) (5x 4) F O I L
35
Multiplying by FOIL
(3x 2) (5x 4) F O I L
36
Multiplying by FOIL
(3x 2) (5x 4) F O I L
15x2 22x 8
37
Multiplying by FOIL
(5x 1) (5x 1) F O I L
38
Multiplying by FOIL
(5x 1) (5x 1) F O I L
39
Multiplying by FOIL
(5x 1) (5x 1) F O I L
40
Multiplying by FOIL
(5x 1) (5x 1) F O I L
41
Multiplying by FOIL
(5x 1) (5x 1) F O I L
42
Multiplying by FOIL
(5x 1) (5x 1) F O I L
43
Multiplying by FOIL
(5x 1) (5x 1) F O I L
25x2 10x 1
44
Multiplying by FOIL
(5x 1)2 (5x 1) (5x 1)
25x2 10x 1
45
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.
• Square of a Binomial
• (a b)2 a2 2ab b2
• (a b)2 a2 2ab b2
• Product of the Sum and Difference of Two Terms
• (a b)(a b) a2 b2

46
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.

47
Evaluating Polynomials

• We can use function notation to represent
  polynomials.
• For example, P(x) 2x3 3x 4.
• Evaluating a polynomial for a particular value
  involves replacing the value for the variable(s)
  involved.

Example
Find the value P(?2) 2x3 3x 4.
2(?2)3 3(?2) 4
P(?2)
2(?8) 6 4
?6
48
Evaluating Polynomials
Techniques of multiplying polynomials are often
useful when evaluating polynomial functions at
polynomial values.
Example
If f(x) 2x2 3x 4, find f(a 3). We replace
the variable x with a 3 in the polynomial
function. f(a 3) 2(a 3)2 3(a 3) 4
2(a2 6a 9) 3a 9 4 2a2 12a 18
3a 9 4 2a2 15a 23
49
Homework

• 5.3 3-45 multiplies odd 71-77 odd
• 5.4 1-33 odd 49, 53, 55, 61, 65

50
SUMMARY

• Can You?...
• 1) multiply monomials and polynomials.
• 2) multiply special types of binomials using the
  FOIL method.

51
The Power Rule

• The Power Rule and Power of a Product or Quotient
  Rule for Exponents
• If a and b are real numbers and m and n are
  integers, then

Power Rule
(am)n amn
(ab)n an bn
Power of a Product
Power of a Quotient
52
The Power Rule
Example
Simplify each of the following expressions.
(23)3
29
512
233
(x4)2
x8
x42
53 (x2)3 y3
125x6 y3
(5x2y)3
53
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
54
Simplifying Expressions
Simplify by writing the following expression with
positive exponents or calculating.
55
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
56
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