5.1&5.2 Exponents

8 2 =8 • 8 = 64 2 4 = 2 • 2 • 2 • 2 = 16 x 2 = x • x Base = x Exponent = 2 x 4 = x • x • x • x Base = x Exponent = 4

Exponents of 1

Anything to the 1 power is itself 5 1 = 5 x 1 = x (xy) 1 = xy

Zero Exponents

Anything to the zero power = 1 5 0 = 1 x 0 = 1 (xy) 0 = 1

Negative Exponents

5 -2 = 1/(5 2 ) = 1/25 x -2 = 1/(x 2 ) xy -3 = x/(y 3 ) (xy) -3 = 1/(xy) 3 = 1/(x 3 y 3 ) a -n = 1/a n 1/a -n = a n a -n /a -m = a m /a n

10 0 10 1 10 2 = 1 = 10 = 100 10 3 = 1000 10 4 = 10000

Powers with Base 10

10 0 10 -1 = 1 = 1/10 1 10 -2 = 1/10 2 10 -3 = 1/10 3 10 -4 = 1/10 4 = 1/10 = 1/100 = .1

= .01

= 1/1000 = .001

= 1/10000 = .0001

The exponent is the same as the The exponent is the same as the number number of 0 ’ s after the 1. 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.

Scientific Notation is of the form: __.

321

___ x 10

-2

(** Note: Only 1 digit to the left of the decimal)

Changing a number from scientific notation to standard form Step 1:

Write the number down without the 10 n part.

Step 2:

Find the decimal point

Step 3: Step 4:

Move the decimal point n places in the ‘ number-line ’ direction of the sign of the exponent.

Fillin any ‘ empty moving spaces ’ with 0.

5.321

.05321

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 10 n .0 5 3 2 1

adding a

x10 -n

where n is the # of digits moved left where n is the #digits moved right

= 5.321 x 10 -2

Raising Quotients to Powers

a n b = a n b n a -n b = a n b n = b n a n = b n a Examples: 3 2 3 2 = 9 = 2 16 2x 3 = (2x) 3 y y = 8x 3 y 3 2x -3 = (2x) -3 = 1 y 3 y -3 (2x) 3 = (2x) 3 = y 3 8x 3

Product Rule

a m • a n = a (m+n) x 3 • x 5 = xxx • xxxxx = x 8 x -3 • x 5 = xxxxx = x 2 xxx 1 = x 2 x 4 y 3 x -3 y 6 = xxxx•yyy•yyyyyy = xy 9 xxx 3x 2 y 4 x -5 • 7x = 3xxyyyy • 7x = 21x -2 y 4 = 21y 4 xxxxx x 2

Quotient Rule

a m a n = a (m-n) 4 3 = 4 • 4 • 4 = 4 1 = 4 4 3 = 64 = 8 4 2 4 • 4 4 2 16 2 = 4 x 5 = xxxxx = x 3 x 5 = x (5-2) x 2 xx x 2 = x 3 15x 5x 4 2 y 3 = 15 xx yyy = 3y y 5 xxxx y x 2 2 15x 2 y 3 = 3 • x -2 • y 2 = 3y 5x 4 y x 2 2 3a -2 b 5 9a 4 b -3 = 3 bbbbb bbb = b 8 3a -2 b 9aaaa aa 3a 6 9a 4 b -3 5 = a (-2-4) b (5-(-3)) = a -6 b 8 = b 8 3 3 3a 6

Powers to Powers

(a m ) n = a mn (a 2 ) 3 a 2 • a 2 • a 2 = aa aa aa = a 6 (2 4 ) -2 = 1 ( 2 = 1 = 1 = 1/256 4)2 2 4 • 2 4 (2 4 ) -2 = 2 -8 = 1 = 1 16 • 16 2 8 256 (x 3 ) -2 = x –6 = x 10 (x -5 ) 2 x –10 x 6 = x 4

Products to Powers

(ab) n = a n b n (6y) 2 = 6 2 y 2 = 36y 2 (2a 2 b -3 ) 2 = 2 2 a 4 b -6 4(ab 3 ) 3 4a 3 b 9 = 4a 4 4a 3 b 9 b 6 = a b 15

What about this problem?

5.2 x 10 3.8 x 10 5 14 = 5.2/3.8 x 10 9



1.37 x 10 9 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 b 2 = a  25 = 5 since 5 2 = 25 A number b is a of a number a if b  8 = 2 since 2 3

cube root

= 8 3 = a Notice that 25 breaks down into 5 • 5 So,  25 =  5 • 5 Notice that 8 breaks down into 2 • 2 • 2 So,  8 =  2 • 2 • 2 Note: See a ‘ group of 2 ’ -> bring it outside the radical (square root sign).

Example:

  200 =  2 • 100 =  2 • 10 • 10 = 10  2 -25 is not a real number since no number multiplied by itself will be negative See a ‘ group of 3 ’ –> bring it outside the radical (the cube root sign)

Example:

3 3  200 =  2 • 100 = 3  2 • 10 • 10 = = 3   2 • 5 • 2 • 5 • 2 2 • 2 • 2 • 5 • 5 = 2 3  25 Note:  3 -8 IS a real number (-2) since -2 • -2 • -2 = -8

5.3 Polynomials

TERM

• a number:

5 •

a variable

X

• a product of numbers and variables raised to powers

5x 2 y 3 p x (-1/2) y -2 z MONOMIAL

-- Terms in which the variables have only nonnegative integer exponents.

-4 5y x 2 5x 2 z 6 -xy 7

A coefficient is the numeric constant in a monomial.

6xy 3 POLYNOMIAL -

A Monomial or a Sum of Monomials:

4x 2

Binomial – A polynomial with 2 Terms

(X + 5)

Trinomial – A polynomial with 3 Terms

+ 5xy – y 2

(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

(2x 2 –3x +7)

+

(3x 2 + 4x – 2) =

5x 2 + x + 5

(5x 2 –6x + 1) – (-5x 2 + 3x – 5) = (5x 2 =

10x 2

–6x + 1) + (5x 2

– 9x + 6

- 3x + 5)

Types of Polynomials

f(x) = 3 f(x) = 5x –3 f(x) = x 2 –2x –1 f(x) = 3x 3 + 2x 2 – 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 1 st polynomial by every term in the 2 nd polynomial

Step 2:

Combine Like Terms

Step 3:

Place in Decreasing Order of Exponent

4x 2 (2x 3 + 10x 2 – 2x – 5)

=

8x 5 + 40x 4 –8x 3 –20x 2 (x + 5) (2x 3 + 10x 2 – 2x – 5)

= 2x 4 + 10x 3 + 10x 3 + 50x 2 – 2x 2 – 5x – 10x – 25 =

2x 4 + 20x 3 + 48x 2 –15x -25

Another Method for Multiplication

Multiply: (x + 5) (2x 3 + 10x 2 – 2x – 5) x 2x 3 2x 4 10x 2 10x 3 – 2x – 5 -2x 2 -5x 5 10x 3 50x 2 -10x -25 Answer: 2x 4 + 20x 3 +48x 2 –15x -25

Binomial Multiplication with FOIL

F.

(First) (2x)(x) 2x 2 2x 2

(2x + 3) (x - 7)

O.

(Outside) (2x)(-7) I.

(Inside) (3)(x) L.

(Last) (3)(-7) -14x 3x -21 -14x + 3x -21

2x 2 - 11x -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: 3x 4 – 9x 3 +12x 2 3x 2 (x 2 – 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: 2x 2 + x = 0 Step 1: Factor Step 2: Zero Product Step 3: Solve for X x (2x + 1) = 0 x = 0 or 2x + 1 = 0 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: x 2 – 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

Factor: 1. y 2 + 7y –30 2.

10x 2 +3x –18

Practice

4. –15a 2 –70a + 120 5. 3m 4 + 6m 3 –27m 2 3. 8k 2 + 34k +35 6. x 2 + 10x + 25

5.7 Special Types of Factoring

Square Minus a Square A 2 – B 2 = (A + B) (A – B) Cube minus Cube and Cube plus a Cube (A 3 – B 3 ) = (A – B) (A 2 + AB + B 2 ) (A 3 + B 3 ) = (A + B) (A 2 - AB + B 2 ) Perfect Squares A 2 + 2AB + B 2 = (A + B) 2 A 2 – 2AB + B 2 = (A – B) 2

5.8 Solving Quadratic Equations

General Form of Quadratic Equation a x 2 + b x + c = 0 a, b, c are real numbers & a



0 A quadratic Equation: x 2 – 7x + 10 = 0 a = __ __ -7 10 Methods & Tools for Solving Quadratic Equations 1.

Factor 2.

3.

Apply zero product principle (If AB = 0 then A = 0 or B = 0) Quadratic Formula (We will do this one later) Example1: x 2 – 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: 4x 2 – 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

x 3 = 4x x 3 - 4x = 0 x (x 2 – 4) = 0 x (x – 2)(x + 2) = 0 x = 0 x – 2 = 0 x + 2 = 0 x = 2 x = -2 2x 3 + 2x 2 - 12x = 0 2x (x 2 + x – 6) = 0 2x (x + 3) (x – 2) = 0 2x = 0 or x + 3 = 0 or x – 2 = 0 x = 0 or x = -3 or x = 2

Solving By Grouping

x 3 – 5x 2 – x + 5 = 0 (x 3 – 5x 2 ) + (-x + 5) = 0 x 2 (x – 5) – 1 (x – 5) = 0 (x – 5)(x 2 – 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 c Hypotenuse Pythagorean Theorem a a 2 + b 2 = c 2 C b A (Leg1) 2 + (Leg2) 2 = (Hypotenuse) 2 Legs