Math 6A ( Periods 2 & 7)

Subtracting Integers 11-3

The life story about someone who was so negative-- you wanted to take a little negativity away but since you can't do that you add a little positiveness-- works in math as well!!
Instead of subtracting ... "ADD THE OPPOSITE!"
We will prove it in class with our little red and yellow tiles... If you need to review, make your own out of red and yellow paper-- or whatever colors you want!!

In life-- to take away a little negative-- add some positive

To take away an integer... add its opposite.

Rule from our textbook
for all integers a and b

a - b = a + (the opposite of b) or
a - b = a + (-b)

Instead of subtracting.. "ADD THE OPPOSITE"
make sure you do the check, check.. you need to have two check marks.. one changing the subtraction to addition and the other changing the sign of the 2nd number to its opposite.

5 - ^- 2=
5 + ⁺ 2= 7

-2 - 5 =
- 2 + ^- 5 =
before I give you the answer look... we are looking at
-2 + ^-5
We are adding two negatives.. so we are back to the rules from Section 11-2...
when adding the same sign just add the number and use their sign so
- 2 + ^- 5 = -7

But what about -2 - ^-5 ?
adding the opposite, we get
-2 + ⁺ 5 .
Now, the signs are different so the rule from Section 11-2 is
ask yourself... "Who wins?" and "By how much?"

Okay, here the positive wins so I know the answer will be +
and by how much means.. to take the difference
5-2 = 3
so -2 + ⁺ 5 = +3
Do you need to put the + sign? No, but I like to in the beginning to show that I checked WHO WON!!

What about
-120 - ^-48?
add the opposite
-120 + ⁺ 48 follows the
Different signs rule... so
ask yourself
Who wins? answer: the negative.. so I know the answer will be negative..
And "By How Much?" take the difference 120-48 = 72
so
-120 + ⁺ 48 = -72
or

-120 - 48
-120 + (-48) = -168

NEVER EVER CHANGE THE FIRST NUMBER'S SIGN!!

WALK THE LINE, the number line-- that is!!
Remember... Attitude is such a little thing... but it makes a BIG difference!!
Always start with a positive attitude!!
When you walk the line, Which way are you always facing when you start???

Always attempt to get everything into addition so we can follow the rules of Section 11-2 Adding Integers.
1) SAME SIGN rule---> just add the numbers and use their sign
-4 + -5 = -9
2) DIFFERENT SIGNS rule
ask yourself those 2 important questions
a) Who wins? (answer is either negative or positive)
b) By how much? (Take the difference)

Algebra Honors ( Period 4)

Adding & Subtracting Polynomials 8-1

VOCABULARY

Monomial: a constant, a variable or product of variables and WHOLE number exponents – one term

Binomial: The SUM of TWO monomials

Trinomial: The SUM of THREE monomials

Polynomial: a monomial or the sum of monomials

Determine whether each expression is a polynomial. If it is a polynomial, find the degree and determine whether it is a monomial, binomial or trinomial.

Example 1:  4y – 5xz 
  Yes 4y and -5xz is really the sum of 4y and -5xz. Degree: 2  binomial

Example 2:  -6.5
Yes, -6.5 is a real number. Degree: 0 monomial

Example 3: 7a^-3 + 9b   No it is not a polynomial because 7a^-3 the exponent is NOT a whole number exponent.

Example 4: 6x³ + 4x + x + 3 Yes  it actually simplifies to 6x³ + 5x  + 3 the Degree is 3 and it is a trinomial

Degree of monomial: The sum of the exponents of all the variables in the term

Degree of polynomial: The highest degree of any term in a polynomial

Although the terms of a polynomial can be written in any order, polynomials in one variable are usually written in standard form.

Standard form  (Descending form): Place variables in alphabetical order with the highest power  first

Leading coefficient: The coefficient of the term that has the highest degree.

Write each polynomial in standard form Identify the leading coefficient

Example 1: 3x² + 4x⁵ -7x
Find the degree of each term
The greatest degree is 5.  so    4x⁵ + 3x² -7x is its standard form , with a leading coefficient of 4

Example 2: 5y -9-2y⁴ -6y³ 
Find the degree of each term
The greatest degree is 4 so,  -2y⁴ -6y³ +5y -9   is its standard form, with a leading coefficient of -2. Notice that sign of the coefficient is kept!

Degree	Name
0	constant
1	linear
2	quadratic
3	cubic
4	quartic
5	quintic
6	6th degree

Add and Subtract Polynomials

Adding polynomials involves adding like terms. You can group by using a horizontal or vertical format.

Find each sum
Example 1; ( 2x² + 5x -7) + (3 – 4x² + 6x)
Horizontal method:
Group and combine like terms

[2x² + (– 4x² )] + [5x + 6x] + {-7 + 3]

– 2x²  + 11x -4

Example 2: (3y + y³ -5) + (4y²- 4y+ 2y³ +8)

Align like terms in columns and combine. Insert a placeholder to help align your terms, if necessary

 y³ + 0y² + 3y -5
+
2y³ + 4y² – 4y + 8

3y³ + 4y²  – y  + 3

You can subtract a polynomial by ADDING its additive inverse. To find the additive invers of a polynomial, write the opposite of each term. That is, distribute the negative to each of the terms.

-(3x² + 2x - 6) = -3x² - 2x + 6

Find each difference

Example 1 (3 – 2x + 2x²) – (4x -5 + 3x²)
Horizontal Method
Subtract 4x -5 + 3x² by ADDING its additive inverse so
( 3 – 2x + 2x²) – (4x -5 + 3x²) = ( 3 – 2x + 2x²) + (-4x +5  -3x²)

Group like terms
[2x²  + (– 3x²)] + [( -2x ) + (-4x)] + [3 + 5]
x² – 6x + 8

Example 2 (7p + 4p³ -8) – (3p² + 2 -9p)
Vertical Method

Align like terms in columns and subtract by adding the additive inverse

4p³ + 0p² + 7p – 8        ADD THE OPPOSITE   4p³ + 0p² + 7p – 8       
(-)     3p² - 9p + 2                                               (+)    - 3p² + 9p - 2
                                                                              4p³ -3p²+ 16p -10

Adding or subtracting integers results in an integers so the set of integers is closed under addition and subtraction. Similarly adding and subtracting polynomials results in a polynomial so the set of polynomials is closed under addition and subtraction.  

Algebra ( Period 5)

Adding and  & Subtracting Polynomials 8-1

VOCABULARY

Monomial: a constant, a variable or product of variables and WHOLE number exponents – one term

Binomial: The SUM of TWO monomials

Trinomial: The SUM of THREE monomials

Polynomial: a monomial or the sum of monomials

Degree of monomial: The sum of the exponents of all the variables in the term

Degree of polynomial: The highest degree of any term in a polynomial

Standard form  (Descending form): Place variables in alphabetical order with the highest power  first

Leading coefficient: The coefficient of the term that has the highest degree.

Degree	Name
0	constant
1	linear
2	quadratic
3	cubic
4	quartic
5	quintic
6	6th degree

Math 6A ( Periods 2 & 7)

Adding Integers 11-2

Rules: The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.

So- if the two numbers have the same sign, use their sign and just add the numbers.

-15 + -13 = - 28

-10 + -4 = -14

Rules: The sum of a positive integer and a negative integer is :

POSITIVE… IF the positive number has a greater absolute value

NEGATIVE… IF the negative number has a greater absolute value

ZERO… IF both numbers have the same absolute value

Think of a game between two teams- The POSITIVE TEAM and The NEGATIVE TEAM.

30 + -16 … ask yourself the all important question…
“WHO WINS?
in this case the positive and then ask
“BY HOW MUCH?”
take the difference 14

14 + - 52…
“WHO WINS?”
the negative… “BY HOW MUCH?”
38
so the answer is -38


(-2 + 3) + - 6 you can work this 2 ways

(-2 + 3) + - 6 = 1 + -6 = -5 or
using all the properties that work for whole numbers
Commutative and Associative properties of addition
can change expression to (-2 + -6) + 3 or -8 + 3 = -5 you still arrive at the same solution.

You want to use these properties when you are adding more than 2 integers.
First look for zero pairs—you can cross them out right away!!
3 + (-3) = 0
-9 + 9 = 0

Then you can use C(+) to move the integers around to make it easier to add them together rather than adding them in the original order. In addition, you can use A(+) to group your positive and negative numbers in ways that make it easier to add as well.

One surefire way is to add all the positives up… and then add all the negatives up.
At this point ask yourself that all important question… WHO WINS? …
use the winner’s sign..
and then ask yourself..
BY HOW MUCH?

example:

-4 + 27 +(-6) + 5 + (-4) + (6) + (-27) + 13

Taking a good scan of the numbers, do you see any zero pairs?
YES—so cross them out and you are left with
-4 + 5 + (-4) + 13
add your positives 5 + 13 = 18
add your negatives and use their sign – 4 + -4 = -8

Okay, Who wins? the positive
By how much? 10
so
-4 + 27 +(-6) + 5 + (-4) + (6) + (-27) + 13 = 10

Math 6A ( Periods 2 & 7)

Integers and Absolute Value 11-1

The following numbers are integers:

… -3, -2, -1, 0, 1, 2, 3,…

The absolute value of an integer is the distance between the number and zero on a number line. The absolute value of a number a is written │a│

Find the absolute value of 2 and graph it on the number line

Find the absolute value of -3 and graph it on a number line

Remember plotting a point involves actually putting a “closed dot” on the number line at the point you are graphing! The numbers go BELOW the number line—not above it

Are absolute values always positive?  What about ZERO?

When you write the notation for the absolute value, it means “take the absolute value of the number inside the symbols”

A number line can be used to compare and order integers.
Numbers to the left are LESS THAN numbers to the right.
Numbers to the right are MORE THAN numbers to the left.

Compare 1 and  │-4│     One way is to graph them on the number line.

The freezing point is the temperature at which a liquid becomes a solid.

Substance	Freezing Point ⁰C
Butter	35
Airplane Fuel	-53
Honey	-3
Mercury	-39
Candle Wax	55

Which substance has the lowest freezing point?

Is the freezing point of mercury or butter closer to the freezing point of water,  0⁰C?

The freezing point of water is  so you can use absolute values to compare
Mercury:  │-39│= 39                       Butter: │35│= 35

Because 35 is less than 39, the freezing point of butter is closer to the freezing point of water.

Good thinking: Determine whether the statement is true of false

A. If x < 0 then  │x│ = -x     ( THIS IS TRUE!... why?)

B. The absolute value of every integer is positive. (THIS IS FALSE… why?)