1

4.1 Exponents

n is your power; x is your base

Read x to the nth power

3 factors of x, 5 factors of y

xxxyyyyy

What is the coefficient for the above term?

What is the power on this term? 3x

nx

53yx

2

Rules of Exponents:

Product Rule:

When multiplying like bases, add the exponents.

1077

1257

xxx

xxx

xxx nmnm

3

Rules of Exponents:

Quotient Rule:

When dividing like bases, subtract the exponents.

1477

257

xxx

xxx

xxx nmnm

4

Rules of Exponents:

Zero Exponent Rule:

Any number raised to the zero power = 1

1300

13

1

0

0

0

x

5

Rules of Exponents:

Power Rule:

When taking a power to a power, multiply exponents.

6222

632 )(

)(

xxxx

xx

xx mnnm

6

Rules of Exponents:

Expanded Power Rule:

When taking a single term to a power, you must take each part to the power.

2222 1644 xxx

yb

xa

by

axmm

mmm

7

Rules of Exponents:

If you forget one of the rules, make up an easy problem that you know the answer to. From this problem, you can reinvent the rule for yourself:

22

221

11

2

211

4)2(

)(

xx

xx

xx

x

xxx

8

Negative

Exponent

Rule

Fraction raised

to a negative

exponent rule mm

mm

a

b

b

a

aa

aa

xx

x

22

22

1

1

0,1

9

4.2 Negative Exponents

Negative Exponent Rule: A negative exponent sends the base to the denominator. If it is already in the denominator, it will actually come back up to the top.

When simplifying, your final answer should have NO negative exponents.

10

4.3 Scientific NotationScientific Notation is a way to write VERY

large or VERY small numbers

Scientific Notation always has TWO parts:

1) A number between 1 and 10

(number must be > 1 AND < 10)

2) A power of ten

11

4.3 Scientific Notation

Examples of numbers and their equivalent scientific notation form:

107,500,000 = 1.075 x 108

0.000756 = 7.56 x 10-4

12

4.3 Scientific NotationTo put a number that is in standard notation

INTO scientific notation,

1) Drop a decimal point in to create a number between 1 and 10

107,500,000 = drop a decimal between 1 and 0 = 1.075

0.000756 = drop a decimal in between 7 and 5 = 7.56

13

4.3 Scientific Notation2) Count the number of places you need to move the

decimal, noting the direction as well. (positive numbers go to the right and negative numbers to the left)

1.075 x 10? How many places do I need to move to get back to 107,500,000?

1.075 x 108

7.56 x 10?

How many places do I need to move to get back to 0.000756?

7.56 x 10-4

14

4.3 Multiplying or Dividing numbers in Scientific Notation

You can not add or subtract numbers when they are in scientific notation, but you can multiply or divide them. And it is not hard either – just multiply or divide the corresponding parts.

(3.6 x 10-2) x (2.0 x 107) = 7.2 x 105

(3.6 x 10-2) / (2.0 x 107) = 1.8 x 10-9

15

4.3 Scientific NotationBe careful! Sometimes your answer will not be in

proper scientific notation: 57 x 106

This is not proper scientific notation because 57 is not between 1 and 10. Move the decimal over and adjust your power: 5.7 x 107

If you move your dec point to the left, you add one to your power of ten.

If you move your dec point to the right, you subtract one from your power of ten.

16

4.4 Addition/Subtraction of Polynomials

A Polynomial is an expression containing the sum of a finite number of terms containing x

3x2 + 5x3 – 5 + 12x7

Polynomials are normally written in descending order of the variable: highest exponent on the variable first and down from there; Constant is always last (5x0)

12x7 + 5x3 + 3x2 – 5

17

4.4 Addition/Subtraction of Polynomials

Polynomial-general term for these expressions

Monomial-1 term 6x

Binomial-2 terms 6x + 8

Trinomial-3 terms 6x2 – 8x + 4

18

4.4 Addition/Subtraction of Polynomials

Degree of a term-exponent of the variable in that term

6x2 – 8x + 4 x2y + x2y3 - x

2 1 0 3 5 1

Degree of a polynomial-is the same as that of its highest-degree term

6x2 – 8x + 4 x2y + x2y3 - x

2nd degree 5th degree

19

4.4 Addition/Subtraction of Polynomials

To add polynomials, combine like terms. Remember that like terms have the same variables and the same degrees of those variables.

To subtract polynomials, use the distributive property to remove parentheses (change every sign in the parenthesis of the polynomials being subtracted) and then combine like terms.

Show columns too.

20

4.5 Multiplication of Polynomials

Monomial x monomial

(4x2)(5x3) = 20x5

Monomial x polynomial (distribute)

4x2 (5x3 +2x2 +5x – 7)=

20x5 +8x4 +20x3 – 28x2

Binomial x binomial (FOIL)

(x+5)(x-7)=x2 - 7x + 5x – 35=x2 – 2x - 35

21

4.5 Multiplication of Polynomials

Difference of Squares

-watch the middle term drop out.

(a+b)(a-b) = a2 + ab – ab – b2 = a2 – b2

(x+4)(x-4) = x2 +4x – 4x – 16 = x2 - 16

Square of binomial formulas

(a+b)2 = (a+b)(a+b)=a2+2ab+b2

(a-b)2 = (a-b)(a-b)=a2-2ab+b2

22

4.5 Multiplication of PolynomialsAny two polynomials can be multiplied together by

distributing each term of the 1st through the 2nd

(x2 + 3x + 7)(4x3 + x2 – 7x – 2)

4x5 + 1x4 – 7x3 – 2x2

12x4 + 3x3 – 21x2 – 6x

28x3 + 7x2 – 49x – 14

4x5 + 13x4 + 24x3 – 16x2 – 55x - 14

23

4.6 Division of Polynomials

Short division:

Divide a polynomial by a monomial-

-divide each term of the polynomial by the monomial

10x2 – 4x 10x2 - 4x 5x - 2

2x 2x 2x

24

4.6 Division of PolynomialsLong division-divide a polynomial by a binomial-(x2 + 3x + 2) ÷ (x + 1)

x + 2x + 1 x2 + 3x + 2

-(x2 + 1x)2x + 2

-(2x + 2)0

No remainder—if you had a remainder, what would you do with it?

25

4.6 Division of Polynomials

Long division-divide a polynomial by a binomial-(x2 + 3x + 7) ÷ (x + 1)

x + 2x + 1 x2 + 3x + 7

-(x2 + 1x)2x + 7

-(2x + 2)5

X + 2 + 1

5

x

26

4.6 Division of Polynomials

To check your answer, multiply the divisor and the quotient (plus the remainder if you have one) and it will equal the dividend

Quotient + Remainder

DivisorDividend

27

4.6 Division of PolynomialsIf you are missing a term when the polynomials are

written in descending order, place a zero where the missing term should go – as a place holder.

(9x2 - 16) ÷ (3x - 4)

3x + 4

3x - 4 9x2 + 0x - 16

-(9x2 - 12x)

12x - 16

-(12x - 16)

0