Transcript + (-x + 5)

5.1&5.2 Exponents
82 =8 • 8 = 64
24 = 2 • 2 • 2 • 2 = 16
x2 = x • x
Base = x
Exponent = 2
x4 = x • x • x • x
Base = x
Exponent = 4
Exponents of 1
Anything to the 1 power is itself
51 = 5
Zero Exponents
Anything to the zero power = 1
x1 = x (xy)1 = xy
50 = 1
x0 = 1
(xy)0 = 1
Negative Exponents
5-2 = 1/(52) = 1/25
x-2 = 1/(x2)
xy-3 = x/(y3)
a-n = 1/an
1/a-n = an
a-n/a-m = am/an
(xy)-3 = 1/(xy)3 = 1/(x3y3)
Powers with Base 10
100 = 1
101 = 10
102 = 100
103 = 1000
104 = 10000
The exponent is the same as the
number of 0’s after the 1.
100 = 1
10-1 = 1/101
10-2 = 1/102
10-3 = 1/103
10-4 = 1/104
= 1/10
= 1/100
= 1/1000
= 1/10000
= .1
= .01
= .001
= .0001
The exponent is the same as the number
of digits after the decimal where 1 is placed
Scientific Notation uses the concept of powers with base 10.
Scientific Notation is of the form: __. ______ x 10
(** Note: Only 1 digit to the left of the decimal)
You can change standard numbers to scientific notation
You can change scientific notation numbers to standard numbers
Scientific Notation
Scientific Notation uses the concept of powers with base 10.
5 ______
321
Scientific Notation is of the form: __.
x 10
(** Note: Only 1 digit to the left of the decimal)
-2
Changing a number from scientific notation to standard form
5.321
Step 1: Write the number down without the 10n part.
Step 2: Find the decimal point
Step 3: Move the decimal point n places in the
‘number-line’ direction of the sign of the exponent.
.05321
Step 4: Fillin any ‘empty moving spaces’ with 0.
Changing a number from standard form to scientific notation
Step1: Locate the decimal point.
Step 2: Move the decimal point so there is 1 digit to the left of the decimal.
Step 3: Write new number adding a x 10n where n is the # of digits moved left
adding a x10-n where n is the #digits moved right
.0 5 3 2 1
= 5.321 x 10-2
Raising Quotients to Powers
a
b
n
Examples:
an
= bn
-n
a
b
2
3
4
32
42
=
2x
y
2x
y
3
=
=
bn
an
= b
a
9
= 16
3
(2x)
= 3 =
y
-3
=
a-n
b-n
(2x)-3
=
y-3
8x3
y3
1
y-3(2x)3
=
y3
y3
(2x)3 = 8x3
n
Product Rule
am • an = a(m+n)
x3 • x5 = xxx • xxxxx = x8
x-3 • x5 = xxxxx = x2 = x2
xxx
1
x4 y3 x-3 y6 = xxxx•yyy•yyyyyy = xy9
xxx
3x2 y4 x-5 • 7x = 3xxyyyy • 7x = 21x-2 y4 = 21y4
xxxxx
x2
Quotient Rule
am = a(m-n)
an
43 = 4 • 4 • 4 = 41 = 4
42
4•4
43 = 64
42
16
x5 = xxxxx = x3
x2
xx
x5 = x(5-2) = x3
x2
15x2y3 = 15 xx yyy = 3y2
5x4y
5 xxxx y
x2
3a-2 b5 = 3 bbbbb bbb = b8
9a4b-3
9aaaa aa
3a6
= 8
2
= 4
15x2y3 = 3 • x -2 • y2 = 3y2
5x4y
x2
3a-2 b5
9a4b-3
= a(-2-4)b(5-(-3)) = a-6 b8 = b8
3
3
3a6
Powers to Powers
(am)n = amn
(a2)3 a2 • a2 • a2 = aa aa aa = a6
(24)-2 = 1
(24)2
(x3)-2 = x –6
(x -5)2 x –10
= 1
=1
= 1/256
24 • 24 16 • 16
= x 10 = x4
x6
(24)-2 = 2-8 = 1 = 1
28 256
Products to Powers
(ab)n = anbn
(6y)2 = 62y2 = 36y2
(2a2b-3)2 = 22a4b-6 = 4a4
= a
4(ab3)3
4a3b9
4a3b9b6
b15
What about this problem?
5.2 x 1014
3.8 x 105
= 5.2/3.8 x 109  1.37 x 109
Do you know how to do exponents on the calculator?
Square Roots & Cube Roots
A number b is a square root
of a number a if b2 = a
A number b is a cube root
of a number a if b3 = a
25 = 5 since 52 = 25
3
Notice that 25 breaks down into 5 • 5
So, 25 =  5 • 5
Notice that 8 breaks down into
3
2 • 2 • 2 So, 8 =  2 • 2 • 2
See a ‘group of 2’ -> bring it outside the
radical (square root sign).
See a ‘group of 3’ –> bring it outside
the radical (the cube root sign)
Example: 200 = 2 • 100
= 2 • 10 • 10
= 10 2
Note: -25 is not a real number since no
number multiplied by itself will be negative
8 = 2 since 23 = 8
3
3
Example: 200 = 2 • 100
3
= 2 • 10 • 10
3
= 2 • 5 • 2 • 5 • 2
3
= 2 • 2 • 2 • 5 • 5
3
= 2 25
3
Note: -8
IS a real number (-2) since
-2 • -2 • -2 = -8
5.3 Polynomials
TERM
• a number:
• a variable
5
X
• a product of numbers and variables raised to powers 5x2 y3 p
x(-1/2)y-2 z
MONOMIAL
-- Terms in which the variables have only nonnegative integer exponents.
-4
5y
x2
5x2z6
-xy7
6xy3
A coefficient is the numeric constant in a monomial.
POLYNOMIAL - A Monomial or a Sum of Monomials: 4x2 + 5xy – y2 (3 Terms)
Binomial – A polynomial with 2 Terms (X + 5)
Trinomial – A polynomial with 3 Terms
DEGREE of a Monomial
– The sum of the exponents of the variables. A constant term has a degree of 0
(unless the term is 0, then degree is undefined).
DEGREE of a Polynomial is the highest monomial degree of the polynomial.
Adding & Subtracting Polynomials
Combine Like Terms
(2x2 –3x +7) + (3x2 + 4x – 2) =
5x2 + x + 5
(5x2 –6x + 1) – (-5x2 + 3x – 5) = (5x2 –6x + 1) + (5x2 - 3x + 5)
= 10x2 – 9x + 6
Types of Polynomials
f(x) = 3
f(x) = 5x –3
f(x) = x2 –2x –1
f(x) = 3x3 + 2x2 – 6
Degree 0
Degree 1
Degree 2
Degree 3
Constant Function
Linear
Quadratic
Cubic
5.4 Multiplication of Polynomials
Step 1: Using the distributive property, multiply every term in the
1st polynomial by every term in the 2nd polynomial
Step 2: Combine Like Terms
Step 3: Place in Decreasing Order of Exponent
4x2 (2x3 + 10x2 – 2x – 5) = 8x5 + 40x4 –8x3 –20x2
(x + 5) (2x3 + 10x2 – 2x – 5) = 2x4 + 10x3 – 2x2 – 5x
+ 10x3 + 50x2 – 10x – 25
= 2x4 + 20x3 + 48x2 –15x -25
Another Method for Multiplication
Multiply: (x + 5) (2x3 + 10x2 – 2x – 5)
x
5
2x3
10x2
– 2x
–5
2x4
10x3
-2x2
-5x
10x3
50x2
-10x
-25
Answer:
2x4 + 20x3 +48x2 –15x -25
Binomial Multiplication with FOIL
(2x + 3) (x - 7)
F.
(First)
O.
(Outside)
I.
(Inside)
L.
(Last)
(2x)(x)
(2x)(-7)
(3)(x)
(3)(-7)
3x
-21
2x2
-14x
2x2
-14x
2x2
+
- 11x
3x
-21
-21
5.5 & 5.6: Review: Factoring
Polynomials
To factor a number such as 10, find
out
‘what times what’ = 10
10 = 5(2)
To factor a polynomial, follow a similar process.
Factor: 3x4 – 9x3 +12x2
3x2 (x2 – 3x + 4)
Another Example:
Factor 2x(x + 1) + 3 (x + 1)
(x + 1)(2x + 3)
Solving Polynomial Equations By
Factoring
Zero Product Property : If AB = 0 then A = 0 or B = 0
Solve the Equation: 2x2 + x = 0
Step 1: Factor
x (2x + 1) = 0
Step 2: Zero Product
x = 0 or
2x + 1 = 0
Step 3: Solve for X
x = 0 or
x= -½
Question: Why are there 2 values for x???
Factoring Trinomials
To factor a trinomial means to find 2 binomials whose product
gives you the trinomial back again.
Consider the expression: x2 – 7x + 10
The factored form is:
(x – 5) (x – 2)
Using FOIL, you can multiply the 2 binomials and
see that the product gives you the original trinomial expression.
How to find the factors of a trinomial:
Step 1: Write down 2 parentheses pairs.
Step 2: Do the FIRSTS
Step3 : Do the SIGNS
Step4: Generate factor pairs for LASTS
Step5: Use trial and error and check with FOIL
Practice
Factor:
1. y2 + 7y –30
2.
10x2 +3x –18
3. 8k2 + 34k +35
4. –15a2 –70a + 120
5. 3m4 + 6m3 –27m2
6. x2 + 10x + 25
5.7 Special Types of Factoring
Square Minus a Square
A2 – B2 = (A + B) (A – B)
Cube minus Cube and Cube plus a Cube
(A3 – B3) = (A – B) (A2 + AB + B2)
(A3 + B3) = (A + B) (A2 - AB + B2)
Perfect Squares
A2 + 2AB + B2 = (A + B)2
A2 – 2AB + B2 = (A – B)2
5.8 Solving Quadratic Equations
General Form of Quadratic Equation
ax2 + bx + c = 0
a, b, c are real numbers & a 0
A quadratic Equation: x2 – 7x + 10 = 0
1
-7 c = ______
10
a = _____
b = _____
Methods & Tools for Solving Quadratic Equations
1. Factor
2. Apply zero product principle (If AB = 0 then A = 0 or B = 0)
3. Quadratic Formula (We will do this one later)
Example1:
x2 – 7x + 10 = 0
(x – 5) (x – 2) = 0
x – 5 = 0 or x – 2 = 0
+5 +5
+2 +2
x = 5
or
x= 2
Example 2:
4x2 – 2x = 0
2x (2x –1) = 0
2x=0 or 2x-1=0
2 2
+1 +1
2x=1
x = 0 or x=1/2
Solving Higher Degree Equations
x3 = 4x
2x3 + 2x2 - 12x = 0
x3 - 4x = 0
x (x2 – 4) = 0
x (x – 2)(x + 2) = 0
2x (x2 + x – 6) = 0
x=0
2x (x + 3) (x – 2) = 0
x–2=0 x+2=0
2x = 0 or
x + 3 = 0 or x – 2 = 0
x=2
x=0
x = -3 or
x = -2
or
x=2
Solving By Grouping
x3 – 5x2 – x + 5 = 0
(x3 – 5x2) + (-x + 5) = 0
x2 (x – 5) – 1 (x – 5) = 0
(x – 5)(x2 – 1) = 0
(x – 5)(x – 1) (x + 1) = 0
x–5=0
or
x-1=0
or
x+1=0
x=5
or
x=1
or
x = -1
Pythagorean Theorem
Right Angle – An angle with a measure of 90°
Right Triangle – A triangle that has a right angle in its interior.
B
a
C
Legs
c
Hypotenuse
Pythagorean Theorem
a2 + b2 = c2
b
A
(Leg1)2 + (Leg2)2 = (Hypotenuse)2