Algebra Honors (Periods 6 & 7)

Differences of Two Squares 5-5 

(a + b)(a-b) = a² - b²
(a+b) is the sum of 2 numbers
(a-b) is the difference of 2 numbers
= ( first#)² - ( 2nd #) ²

( y -7)( y + 7) = y² - 49
We did the box method to prove this.

(4s + 5t) (4s - 5t)
16s² - 25t²

(7p + 5q)(7p-5q) = 49p² - 25q²
But then we looks at
(7p+5q)(7p+5q) that isn't the difference of two squares that is
49p² + 70pq + 25q²
So let's look at the difference of TWO Squares:

b² -36
So that is ( b + 6)(b -6)

m² - 25
(m + 5)(m -5)

64u² - 25v²
(8u + 5v)(8u -5v)

1 - 16a²

(1+4a)(1- 4a)

But what about 1- 16a⁴

( 1 + 4a²)(1 - 4a²) but we are NOT finished factoring because
(1 - 4a²) is still a difference of two squares so it becomes

( 1 + 4a²)(1 + 2a)(1 - 2a)

t⁵ - 20t³ + 64t

Factor out the GCF first

t(t⁴ - 20t² + 64)
t(t² -16)(t² -4)
YIKES... we have two Difference of Two Squares here...

t(t +4)(t-4)(t +2)(t -2)



81n² - 121
(9n +11)(9n -11)
3n⁵ - 48 n³


Factor out the GCF

3n³ (n² - 16)
3n³(n +4)(n-4)

50r⁸ - 32 r²

Factor out the GCF
2r²(25r⁶ - 16)

2r²(5r³ +4)(5³ -4)

u² - ( u -5) ²
think a² - b ² = (a + b)(a -b)

so
u² - ( u -5) ² =
[u + (u-5)][u - (u-5)]
(2u -5)(5)
= 5(2u-5)

t² - (t-1)²

[t +( t+1)]{t-(t-1)]
2t-1(+1)
=2t -1


What about x²ⁿ - y ⁶ where n is a positive integer

well that really equals
(xⁿ)² - (y³)²
so
(xⁿ + y³)(xⁿ - y³)

x²ⁿ - 25
(xⁿ + 5)(xⁿ - 5)

a⁴ⁿ - 81b⁴ⁿ
(a²ⁿ + 9b²ⁿ)(a²ⁿ - 9b²ⁿ)
= (a²ⁿ + 9b²ⁿ)(aⁿ + 3bⁿ)(aⁿ - 3bⁿ)


When multiplying to numbers such as (57)(63)
think
(60-3)(60 +3)
then the problem becomes so much easier

3600 - 9 = 3591 DONE!!!

(53)(47) = (50 +3)( 50-3)
2500 - 9 = 2491

Math 6A (Periods 1 & 2)

Dividing Decimals 3-9

According to our textbook-
In using the division process to divide a decimal by a counting number, place the decimal point in the quotient directly over the decimal point in the dividend.

Check out our textbook for some examples!!

When a division does not terminate-- or does not come out evenly-- we usually round to a specified number of decimal places. This is done by adding zeros to the end of the dividend, which as you know, does NOT change the value of the decimal. We then divide ONE place beyond the specified number of places.

Divide 2.745 by 8 to the nearest thousandths.
See the set up in our textbook on page 89. Notice that they have added a zero and the end of the dividend ( 2.745 becomes 2.7450) because you want to round to the thousandths and we need to go ONE place additional.
DIVIDE carefully!!

the quotient is 0.3431 which rounds to 0.343


To divide one decimal by another

Multiply the dividend and the divisor by a power of ten that makes the DIVISOR a counting number


Divide the new dividend by the new divisor

Check by multiplying the quotient and the divisor.

Math 7 (Period 4)

Dividing Integers  3.6 cont'd

Averages
Adding  all the number and dividing by the number of items

Make sure that if ZERO is one of the items that you COUNT it as an item!!

Multiplying and Dividing with VARIABLES, INTEGERS, and POWERS

Always plug in a negative number in ( )

Powers of Negative Numbers

The ODD/EVEN Rule

If there is a negative inside parentheses:

Odd number of negative signs or odd power = negative

Even number of negative signs or even power = positive

 Examples

(-2)⁵ = (-2)(-2)(-2)(-2)(-2) = -32

(-2)⁴ = (-2)(-2)(-2)(-2) =  16

If there is a negative BUT no (parenetheses) --> it is always negative

-2⁵ = -32

-2⁴ = -16

So NOW you know why

(-2)⁴  ≠  -2⁴

Again whenever you plug a Negative for a variable ALWAYS put the negative in (   ).
I actually like students to ALWAYS use (  ) when substituting in for ANY number…

Examples

solve:

x² when x = -3

(-3)² = 9

  -x² when x = -3

-(-3)² = -9

10- x² when x = -3

10 - (-3)² = 10 -9 = 1

Algebra Honors ( Periods 6 & 7)

Multiplying Binomials Mentally 5-4

Look at
(3x - 4)(2x+5)

remember FOIL
First terms (3x)(2x)
Outer terms (3x)(5)
Inner Terms (-4)(2x)
Last Terms (-4)(5)

6x² + 15x -8x -20
6x² + 7x - 20
Or use the box method as we have done in class
This is a quadratic polynomial
The quadratic term is a term of degree two

Remember a linear term has a term of degree 1 such as y = 3x + 5

6x² + 7x - 20

The 6x² is the quadratic term
the +7x is the linear term
and the - 20 is the constant term

(x +1)(x +3) = x² + 4x + 3

(y + 2)( y + 5) = y² + 7y + 10
( t -2)( t -3) = t² -5t + 6

( u -4)(u -1) = u² - 5u + 4

What about
( u-4)(u +1) = u ² -3u -4

See the difference between the two?

(7 - k)(4 -k)

28 - 11k + k²


r + 3)(5 - 5)
r² - 25 - 15



(3x - 5y)(4x + y)
12x² - 17xy - 5y²

a + 2b)(a-b)

careful....
a² + ab - 2b²

n(n-3)(2n+1)
first distribute the n
(n² -3n)(2n +1)
2n³ - 5n² - 3n

Solve for
(x-4)(x +9) = (x +5)(x -3)
x² + 5x - 36 = x² + 2x -15
5x - 36 = 2x - 15
3x = 21

x = 7
or in solution set notation {7}

Algebra Honors ( Periods 6 & 7)

Monomial Factors of Polynomial 5-3

In the first chapters we reflected that if a, b, and c, were real numbers and c was not equal to 0,
then
(a + b)/c = a/c + b/c
It also applies to monomials
(5m + 35)/ 5 = 5m/5 + 35/5... which simplifies to  m + 7

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial and then add the results.
From this point on, our textbook has us assume that not divisor equals 0

Divide the following
26uv - 39v
    13v

You can separate each part so that you have
(26uv)/13v  - 39v/13v
which simplifies to
2n -3

Divide

3x⁴ - 9x³y + 6x²y² 
-3x²

This time you notice that -3x² is a factor of all three terms of this polynomial
and you can separate the polynomial into three separate terms or... as taught in class you can easily do each ( carefully)
so that you simplify the fraction to
-x² + 3xy -2y²

Divide:
x³y - 4y + 6x
     xy

Here you might want to show the three terms to see what is happening to each individual term...

x³y 
  xy

-4y
xy

+6x
xy

which simplifies to

x² -4/y + 6/x

You definitely could simplify by crossing out the common factors  each term shares with the divisor.

One polynomial is evenly divisible or just divisible by another polynomial if the quotient is also a polynomial.
So the first two examples show divisibility but this last one does NOT.

You factor a polynomial by expressing it as a product of other polynomials. The factor set for a polynomial have integral coefficients.
You can use division to test for factors!

The greatest monomial factor of a polynomial is the GCF of its terms!

Factor
5x² + 10x
The greatest monomial factor is 5x
You don't want to change the value of your polynomial-- you just want to factor it!
Pull out the 5x and divide by 5x as well.. because you are NOT CHANGING the VALUE
Its simply using the Distributive Property

5x(5x² + 10x)
     5x

5x(x + 2)
To check-- just multiply out using your knowledge of the distributive property!!

Factor
4x - 6x³ + 14x

The greatest monomial factor is 2x

2x(4x - 6x³ + 14x)
   2x
2x(2x - 3x² + 7)

Factor
8a²bc² - 12ab²c²
The greatest monomial factor is 4abc²


4abc²(8a²bc² - 12ab²c²)
  4abc²

4abc²(2a-3b)

Practice these and you will be able too do the division steps mentally.
Check your factorization by multiplying the resulting factors!
Make sure you end up with where you started-- when you check!!

Math 7 ( Period 4)

Multiplying and Dividing Integers 3.5 & 3.6

They have the same rules! (The rules are different from adding and subtracting integers… so keep these straight!)

Textbook Rules:
If you have 2 signs that are the SAME, the product or quotient is POSTIIVE

IF you have 2 signs that are DIFFERENT, the product or quotient is NEGATIVE

Good Guy/ Bad Guy Rules… or Life’s Lessons…

Good thing happens to Good person à that’s Good

Bad thing happens to a Good Personà that’s Bad (That’s not right! We hate when that happens)

Good thing happens to a Bad person à That’s Bad (We hate when good things happen to people who don’t deserve it.

Bad thing happens to a Bad person à that’s good (That’s Karma … they got what they deserved)

+ ∙ + = +

- ∙ + = -

+ ∙ - = -

-  ∙ - = +

It works with division as well!

Finger Rules:

If your index finger represents the negative sign then if  you have two negatives you have the index fingers of both your left hand and your right hand and they make a plus sign!

If you have just one negative it just stays negative because you don’t have another finger to cross it

If you have more than two negatives you just keep using your index fingers to determine the sign.

We did this in class. It’s fun… but once you get it.. you won’t need to keep doing it ( unless you want to keep having fun!)

What if there are more than 2 signs?

Use Aunt Sally’s  Rules (O³) and go left to right

or be a Sign Counter

…an ODD number of negatives = Negative

…an EVEN number of negatives = Positive

Only count the negative signs… you do not worry about how many numbers are involved nor the positives… only count the Negatives

(-2)(-5)(-3) = -30 (there are 3 negatives—and 3 is odd)

(2)(-5)(-3) = +30 (there are 2 negatives and 2 is an even number)

(2)(5)(-3) = -30 (there is only 1 negative and that is odd!)

Math 6A (Periods 1 & 2)

Multiplying or Dividing by a Power of Ten 3-7

We have learned that in a decimal or a whole number each place value is ten times the place value to its right.

10 ∙ 1 = 10
10 ∙ 10 = 100
10 ∙ 100 = 1000

10 ∙ 0.1 = 1
10 ∙ 0.01 = 0.1
10 ∙ 0.001 = 0.01

Notice that multiplying by ten has resulted in the decimal point being moved one place to the right and in zeros being inserted or dropped.

Multiplying by ten moves the decimal point one place to the right

10 ∙ 762 = 7620

762 X 10 = 7620

4931 X 10 = 49,310


10⁴ = 10⋅10⋅10⋅10 = 10,000

2.63874 X 10⁴ = 26,387.4

To multiply a number by the nth power of ten--> move the decimal n places to the right.

0.0047 multiply by 100 = 0.47
0.0047 multiply by 1000 = 4.7

3.1 ÷ 10⁴ = 0.00031


10 ∙ 4.931 = 49.31

At the beginning of this chapter you learned about powers of ten

10⁴ = 10 ∙10 ∙ 10 ∙10 = 10,000

We can see that multiplying by a power of 10 is the same as multiplying by 10 repeatedly.

2.64874 ∙10⁴ = 26,387.4

Notice that we have moved the decimal point four places to the right.

Rule

To multiply a number by the n^th power of ten, move the decimal point n places to the right.



When we move a decimal point to the left, we are actually dividing by a power of ten.


Notice that in dividing by a power of 10 we move the decimal point to the left the same number of places as the exponent. Sometimes we may have to add zeros

Rule

To divide a number by the nth power of ten, move the decimal point n places to the left, adding zeros as necessary.

2386 ÷ 10³ = 2.386

Powers of ten provide a convenient way to write very large numbers. Numbers that are expressed as products of two factors

(1) a number greater than or equal to 1, but less than 10,

AND

(2) a power of ten

are said to be written in scientific notation.

We can write 'a number greater than or equal to 1, but less than 10' as an mathematical inequality 1 ≤ n < 10 To write a number in scientific notation we move the decimal point to the left until the resulting number is between 1 and 10. We then multiply this number by the power of 10, whose exponent is equal to the number of places we moved the decimal point. 4,592,000,000 in scientific notation First move the decimal point to the left to get a number between 1 and 10 4,592,000,000 the first factor in scientific notation becomes 4.592 Since the decimal point was moved 9 places, we multiply 4.592 by 10⁹ to express the number in scientific notation



4.592 x 10⁹ (Yes, you get to use the × symbol for multiplication .. but only for this!!



This is a way to write very large numbers AND very small numbers

Numbers expressed as products of a number greater than or equal to 1 BUT less than 10, AND a power of ten are called Scientific Notation.

Two Factors
(1) 1≤ n < 10
(2) Power of 10
4,592,000,000 becomes 4.592 X 10⁹
moved the decimal 9 places so we must multiply our number by a power of 10⁹

98,000,000 = 9.8 X 10⁷

320,000 = 3.2 X 10⁵

What if I give you 7.04 X 10⁸ and ask you to put it back into STANDARD NOTATION:

704,000,000.

0.0031 = 3.1 X 10^-3
It isn't a negative number its just a very tiny number

1≤ n < 10 0.16 becomes 1.6 x 10 ^-1

Math 7 (Period 4)

Chapter 3.4 Subtracting Integers
NEVER SUBTRACT—Add the Opposite

I call it "Double Check method" because you always change 2 signs

1) Change the subtraction sign to an addition sign (check)

2) Change the subtrahend’s sign (the 2^nd number’s sign) to its opposite
(if it was negative change it to positive,
if it had no sign then put a negative because no sign meant it was positive) (double check)

3) Follow the rules of integer addition from  Section 3.3

Example

5 –(-10) = 5 +(+10) =15

-5 –(-10) = -5 +(+10) = +5

-5- 10 = -5 +(-10) = -15

More than 1 subtrahend?  Double check each one!

By the way NEVER EVER EVER CHANGE THE FIRST NUMBER”S SIGN!!

Identifying terms:

Terms are separated by ADDITION
(Remember there is no such thing as subtraction)

2xy is only ONE term—but it has 3 factors

2xy + 3  is 2 two terms

2xy – 3z- (-10) is 3 terms made up of the following

2xy, -3z, and +10
(you must put the terms in Addition Format to determine their signs)

ALWAYS SIMPLIFY BEFORE EVALUTATING

11x + 14 – 21x + 6 + 12x  when x = 5

You will get the same answer if you plug and chug x = 5 in all of the terms as when you SIMPLIFY FIRST and then plug and chug just once.

11(5) + 14 – 21(5) + 6 12(5)

55 + 14 -105 + 6 + 60

135 + (-105)

30

VS

(11x -21x + 12x) + ( 14 + 6)

2x + 20

2(5) + 20

10 + 20

30

Which way do you want to do these types of problems?

Simplifying first is usually the least work!

Quick review of Coefficient and Constants

Remember that a coefficient goes along with a variable and EVERY VRIABLE MUST HAVE  COEFFICIENT

so

2a – 3b –(-c) –d – 12

has 5 terms, 4 variables, 4 coefficeints and 1 constant

The coefficients are 2, 3, 1, -1

The IDENTITY Property of Multiplication (IDx) says that you can sneak in the “1” by multiplication in front of any variable that has no other coefficient.  

Fractional Coefficients

2x

3

can be written 2/3(x)

2 x

3

So the Coefficient of this one term is 2/3

Review of Absolute Value with subtraction inside

Absolute value symbols are similar to parentheses  in that you must simplify inside using Order of Operations (O³) BEFORE applying the absolute value at the end

│2∙3²-30│

You need to do the power first, then multiply, then subtract and THEN absolute value at the very end.

│2∙9-30│= │18-30│=│-12│ = 12

Math 6 A (Periods 1 & 2)

Adding & Subtracting Decimals 3-6

Rules

1) Write the given numbers one above the other with the decimal points in line

I call that “Stack’ Em”

2) Add any zeros to get the same number of decimal places and then add or subtract ( +/-) as if the numbers were whole numbers.

We put the added zeros in colored pencil to distinguish them!

3) Place the decimal point in the number for the sum or difference in position directly under the decimal points in the given numbers.

add 6.47 + 3.40.8 + 73.523

Stack them   Lining up the decimals

420.793

Subtract 13.94 – 7.693

Again, stack and align the decimal points

6.247

Chelsea (Lindsey) had $ 317.58. She made three purchases of $19.95, $ 27.49, and $ 89.98

How much did she have left?

Add the purchase first… stack them

137.42

Take the total of her purchases and subtract that from the amount she started with $ 317.58

 

180.16

She had $180.16 left!

Estimations-  round to the highest place value of the SMALLEST number.

8.574 + 81.03 + 59.432

First figure out which is the smallest number?

In this case it is 8.574

Round that to its highest place value

8.574 rounded  to the ONES place is 9

so round all the other numbers to the ONES places

and add them

9 + 81 + 59

You can stack them and add normally

9 + 81 + 59 = 149

76.061 – 3.211

when you estimate you get 76  - 3= 73

When you actually add them you arrived at

76.061
-3.211

72.850

Which we write as 72.85

What happens with

(75.004 – 1.32) + ( 41.13 – 2.891)

First take the difference of each set of ( )

then add the differences

73.684 + 38.239 = 111.923

Math 7 (Period 4)

Using Rules to Add Integers 3.3

To add two integers with the same sign à just add them and use their sign

-          7 + (-5) = -12

To add two integers with different signs find the absolute values and take the difference. Use the sign of the number with the larger absolute value.

In 7^th grader terms:

Thinkà  Two teams:  the negatives and the positives

Ask yourself.. Who wins?  (that tells you who has the greatest absolute value)..

Stack them with the winner on top and take the difference ( subtract)

Use the sign of the winner!

The two questions to ask yourself ...Who wins? and By how much? 

43 + (-152)

The negatives win here

stack

152

-43

109

The solution must be -109

What if there are several addends? Here is a great strategy to follow:

1) See if there are any additive inverse first. Cross them out  using Inv+

2) Add the positives to the positives

AND

the negatives to the negatives

What properties allow us to do that? C+ and A+

3) Finally add the positive sum to the negative sum…
See “Who wins?” and “By how much?”

We tried this with a string of addends:

+ 3 + (-2) + 17 + 20 + (-3) + (-17)

We noticed that 3 and (-3) were additive inverses as well as

17 and (-17)

We crossed them out and were left with

(-2) + 20

That became easy—The signs were different so we asked ourselves.. “Who wins? and “By how much?”

the positive won.. by 18 so the answer was

+18

Then we tried:

+4 + (-5) + 18 + 3+ 25 + (-18) +(-4) + (-6)

we noticed we could cross out the +4 and the (-4)

as well as the +18 and the *-18)

Those were both additive inverses.

We were left with

+(-5) + 3+ 35 + (-6)

Add the positives 3 + 25 = 28

Add the negatives +(-5)+(-6) = -11

Then take the difference

28 -11 = 17

Much easier than working from left to right. You will make less “silly” mistakes using these strategies!

Adding Integers with variable expressions

Just substitute in for the variable, putting the substituted number into hugs (  )

Then evaluate using the integer rules… Plug & Chug

y + 5  where y = -12

(-12) + 5 = -7