5.1&5.2 Exponents82 =8 • 8 = 64 24 = 2 • 2 • 2 • 2 = 16

x2 = x • x x4 = x • x • x • x Base = x Base = xExponent = 2 Exponent = 4

Exponents of 1 Zero ExponentsAnything to the 1 power is itself Anything to the zero power = 1

51 = 5 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) (xy)-3 = 1/(xy)3 = 1/(x3y3)

a-n = 1/an 1/a-n = an a-n/a-m = am/an

Powers with Base 10100 = 1101 = 10102 = 100103 = 1000104 = 10000

The exponent is the same as the The exponent is the same as the numbernumber of 0’s after the 1. of digits after the decimal where 1 is placed

100 = 110-1 = 1/101 = 1/10 = .110-2 = 1/102 = 1/100 = .0110-3 = 1/103 = 1/1000 = .00110-4 = 1/104 = 1/10000 = .0001

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 notationYou can change scientific notation numbers to standard numbers

Scientific NotationScientific 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)

-25 321

Changing a number from scientific notation to standard formStep 1: Write the number down without the 10n part.Step 2: Find the decimal pointStep 3: Move the decimal point n places in the ‘number-line’ direction of the sign of the exponent.Step 4: Fillin any ‘empty moving spaces’ with 0.

Changing a number from standard form to scientific notationStep1: 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

5.321

.05321

.0 5 3 2 1= 5.321 x 10-2

Raising Quotients to Powers

a n

b = an

bna -n b

= a-n

b-n= bn

an= b n

a

Examples: 3 2 32 94 42 16= =

2x 3 (2x)3 8x3

y y3 y3= =

2x -3 (2x)-3 1 y3 y3

y y-3 y-3(2x)3 (2x)3 8x3= = = =

Product Ruleam • 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 43 = 64 = 8 = 442 4 • 4 42 16 2

x5 = xxxxx = x3 x5 = x(5-2) = x3

x2 xx x2

15x2y3 = 15 xx yyy = 3y2 15x2y3 = 3 • x -2 • y2 = 3y2 5x4y 5 xxxx y x2 5x4y x2

3a-2 b5 = 3 bbbbb bbb = b8 3a-2 b5 = a(-2-4)b(5-(-3)) = a-6 b8 = b8

9a4b-3 9aaaa aa 3a6 9a4b-3 3 3 3a6

Powers to Powers

(am)n = amn

(a2)3 a2 • a2 • a2 = aa aa aa = a6

(24)-2 = 1 = 1 = 1 = 1/256 (24)2 24 • 24 16 • 16 28 256

(x3)-2 = x –6 = x 10 = x4

(x -5)2 x –10 x 6

(24)-2 = 2-8 = 1 = 1

Products to Powers

(ab)n = anbn

(6y)2 = 62y2 = 36y2

(2a2b-3)2 = 22a4b-6 = 4a4 = a4(ab3)3 4a3b9 4a3b9b6 b15

What about this problem?

5.2 x 1014 = 5.2/3.8 x 109 1.37 x 109

3.8 x 105

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

25 = 5 since 52 = 25

Notice that 25 breaks down into 5 • 5So, 25 = 5 • 5

See a ‘group of 2’ -> bring it outside theradical (square root sign).

Example: 200 = 2 • 100 = 2 • 10 • 10 = 10 2

A number b is a cube root of a number a if b3 = a

8 = 2 since 23 = 8

Notice that 8 breaks down into 2 • 2 • 2 So, 8 = 2 • 2 • 2

See a ‘group of 3’ –> bring it outsidethe radical (the cube root sign)

Example: 200 = 2 • 100 = 2 • 10 • 10 = 2 • 5 • 2 • 5 • 2

= 2 • 2 • 2 • 5 • 5 = 2 25

3

3

3 3

3

3

3

3

Note: -25 is not a real number since nonumber multiplied by itself will be negative

Note: -8 IS a real number (-2) since-2 • -2 • -2 = -8

3

5.3 PolynomialsTERM • a number: 5 • a variable 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.

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.

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

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 Polynomialsf(x) = 3 Degree 0 Constant Functionf(x) = 5x –3 Degree 1 Linear f(x) = x2 –2x –1 Degree 2 Quadraticf(x) = 3x3 + 2x2 – 6 Degree 3 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 TermsStep 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)

2x3 10x2 – 2x – 5

x

5

2x4 10x3 -2x2 -5x

10x3 50x2 -10x -25

Answer: 2x4 + 20x3 +48x2 –15x -25

Binomial Multiplication with FOIL

(2x + 3) (x - 7)

F. O. I. L.(First) (Outside) (Inside) (Last)

(2x)(x) (2x)(-7) (3)(x) (3)(-7)

2x2 -14x 3x -21

2x2 -14x + 3x -21

2x2 - 11x -21

5.5 & 5.6: Review: Factoring Polynomials

To factor a polynomial, follow a similar process.

Factor: 3x4 – 9x3 +12x2

3x2 (x2 – 3x + 4)

To factor a number such as 10, find out ‘what times what’ = 10

10 = 5(2)

Another Example:Factor 2x(x + 1) + 3 (x + 1)

(x + 1)(2x + 3)

Solving Polynomial Equations By Factoring

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

Zero Product Property : If AB = 0 then A = 0 or B = 0

Question: Why are there 2 values for x???

Factoring Trinomials

To factor a trinomial means to find 2 binomials whose productgives you the trinomial back again.

Consider the expression: x2 – 7x + 10

(x – 5) (x – 2)The factored form is:

Using FOIL, you can multiply the 2 binomials andsee 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 FIRSTSStep3 : Do the SIGNSStep4: Generate factor pairs for LASTSStep5: Use trial and error and check with FOIL

Practice

Factor:

1. y2 + 7y –30 4. –15a2 –70a + 120

2. 10x2 +3x –18 5. 3m4 + 6m3 –27m2

1. 8k2 + 34k +35 6. x2 + 10x + 25

5.7 Special Types of FactoringSquare 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 a = _____ b = _____ c = ______

Methods & Tools for Solving Quadratic Equations• Factor • Apply zero product principle (If AB = 0 then A = 0 or B = 0)• Quadratic Formula (We will do this one later)

Example1: Example 2:x2 – 7x + 10 = 0 4x2 – 2x = 0(x – 5) (x – 2) = 0 2x (2x –1) = 0x – 5 = 0 or x – 2 = 0 2x=0 or 2x-1=0 + 5 + 5 + 2 + 2 2 2 +1 +1

2x=1x = 5 or x = 2 x = 0 or x=1/2

1 -7 10

Solving Higher Degree Equations

x3 = 4x

x3 - 4x = 0x (x2 – 4) = 0x (x – 2)(x + 2) = 0

x = 0 x – 2 = 0 x + 2 = 0

x = 2 x = -2

2x3 + 2x2 - 12x = 0

2x (x2 + 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

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.

Legs

Hypotenuse

C A

B

a

b

cPythagorean Theorem

a2 + b2 = c2

(Leg1)2 + (Leg2)2 = (Hypotenuse)2