Math 6A ( Periods 1 & 2)

Decimals 3-3

Although decimals ( termed decimal fractions) had been used for centuries, Simon Stevin in the 16th century began using them on a daily basis and he helped establish their use in the fields of sciences and engineering.

Note that
1/10 = 1/10¹
1/100 = 1/10²
1/1000 = 1/10³

We also know that
1/10= 0/1
1/100 = 0.01
1/1000 = 0.001
1/10000 = 0.0001
and so on... these strings of digits are called decimals.


SO 1/10 = 1/10¹= 0.01 and it is equal to 10^-1
Notice that 10^-1 is NOT a negative number-- it is a small number
and 10^-21 is not a negative number it is a VERY TINY number

AS with whole numbers, decimals use place values. These place values are to the RIGHT of the decimal point.
We need to be able to write decimals in words as well as expanded notation.
In class we used 0.6394 as our example

zero and six thousand three hundred ninety-four ten-thousandths.

Notice how this number when written in words begins...with "ZERO AND"
Why do we need to do that?

Also notice that there is a hyphen between ten and thousandths in ten-thousandths. It is critical to understand when you must place a hyphen.
We read the entire number to the right of the decimal point as if it represented a whole number, and then we give the place value of the digit farthest to the right.

So, although 0.400 is equivalent to 0.4
we must read 0.400 as "zero and four hundred thousandths."

Now look at the following words
"zero and four hundred-thousandths." What is the subtle difference between those two phrases above?
There is a hyphen in the last phrase-- which means that the hundred and the thousandths are attached and represent a place value so

zero and four hundred-thousandths is 0.00004 while
zero and four hundred thousandths is 0.400

Carefully see the distinction!!

Getting back to our 0.6394

to write it in decimals sums and then in exponents:
0 + 0.6 + 0.03 + 0.009 + 0.0004

0 + 6(0.1) + 3(0.01) +9(0.001) + 4(0.0001)

0(10⁰) + 6(10^-1)+ 3(10^-2)+ 9(10^-3)+ 4(10^-4)

14.35 is read as fourteen AND thirty-five hundredths.
When reading numbers, only use the AND to indicate the decimal point

Algebra Honors (Periods 6 & &)

Rate-Time- Distance Problems 4-8

D = rt

Uniform Motion

Three types of problems:

• Motion in opposite direction
• Motion in same direction
• Round Trip

Motion in opposite direction
For this we used different students bicycling ... Jake and  Jordan in 6th period and Henri and Andy from 7th period.

Last year's examples follow:
They start at noon ;60 km apart riding toward each other. They meet at 1:30 PM. If Jake's speed is 4 km/h faster than Jordan's ( Henri is greater by the same from Andy's rate) What are their speeds?

The chart below was take from last year's examples

We set up a chart

Motion in Same Direction

Next we had a fictitious story about Ritika's Helicopter and Maya's plane ( or  Ryan's helicopter and Allison's Smiling Plane) taking off from Camarillo Airport flying north. The helicopter flies at a speed of 180 mi/hr. 20 minutes later the plane takes off in the same direction going 330 mi/hr. How long will it take Maya (or Allison) to over take  Ritika's ( or Ryan's) helicopter?

Let t = plane's flying time

Make sure to convert the 20 minutes ---> 1/3 hours.
We set up a chart .. Here is the chart with last year's names:

When the plane over takes the helicopter they have traveled the exact same distance so set them equal
180(t + 1/3) = 330t

180 t + 60 = 330t
60 = 150t

t = 2/5
which means 2/5 hour. or 24 minutes.

Round Trip

A ski lift carries Jenna ( or Brendan) up the slope at 6 km/h Jenna or Brendan snowboard down 34 km/h. The round trip takes 30 minutes.

Did you see the picture?

Let t = time down
then set up a chart

6(.5 -t) = 34 t
3 - 6t = 34 t
3/40 = t
Now, what's that?

0.75 hr or 4.5 minutes

How far did they snowboard... plug it in
34(0.075) = 2.55 km

Algebra Honors (Periods 6 & 7)

Transforming Formulas 4-7

Formulas are used throughout real life applications-- the book gives an example of the formula for the total piston displacement of an auto engine... we discussed a number of formulas that students recalled such as the following:
A =lw
d = rt
I = Prt
A = ∏r²
A = P(1 + rt)
C = 5/9(F - 32)
y = mx + b
A = bh
C = ∏d
F = Ma
A = ½(b₁b₂h
E= mc<sup>2
A2 + B2 = C2
and even the quadratic formula-- which we will study later this year..



b = ax ; x
just divide both sides by a
b/a = x

Solve P = 2L + 2W for the width, w
P-2L = w
2

C = 5/9(F - 32) Solve for F
We need to multiply both sides by the reciprocal of 5/9
(9/5)C = F - 32
now add 32 to both sides
(9/5)C + 32 = F


Next we tackled</sup> 

We also discussed the restrictions and found that the denominator could not equal zero.


S = v/r ( solving for r) became one of the homework problems that caused some discussion-- until students realized that they had actually found the reciprocal of r or 1/r instead of solving for r!!
We discussed how to solve that dilemma.

Math 6A ( Periods 1 & 2)

Exponents and Powers of Ten 3-1
When two or more numbers are multiplied together--each of the numbers is called a factor of the product.

A product in which each factor is the SAME is called a power of that factor.

2 X 2 X 2 X 2 = 16. 16 is called the fourth power of 2 and we can write this as
2⁴ = 16

The small numeral (in this case the 4) is called the exponent and represents the number of times 2 is a factor of 16.
The number two, in this case, is called the base.

When you are asked to evaluate... simplify... solve... find the answer
That is,
Evaluate
4³ = 4 X 4 X 4 = 16 X 4 = 64

The second and third powers of a numeral have special names.
The second power is called the square of the number and the third power is called the cube.

We read 12² as "twelve squared" and to evaluate it
12² = 12 X 12 = 144

Powers of TEN are important in our number system.
Make sure to check out the blue sheet and glue it into your spiral notebook
First Power: 10¹ but the exponent is invisible = 10
Second Power: 10² = 10 X 10 = 100
Third Power 10³ = 10 X 10 X 10 = 1000
Fourth Power 10⁴ =10 X 10 X 10 X 10 = 10,000
Fifth Power 10⁵ = 10 X 10 X 10 X 10 X 10 = 100,000

Take a look at this list carefully and you will probably see a pattern that we can turn into a general rule:

The exponent in a POWER of TEN is the same as the number of ZEROS when the number is written out.

The number of ZEROS in the product of POWERS OF TEN is the sum of the numbers of ZEROS in the factors.

For example Multiply.
100 X 1000
Since there are 2 Zeros in 100 and 3 zeros in 1000,
the product will have 2 + 3 , or 5 zeroes.
100 X 1000 = 100,000

When you need to multiply other bases:

first multiply each
For example

3⁴ X 2 ³ would be
(3 X 3 X 3X 3) X ( 2 X 2 X 2)
= 81 X 8 = 648

What happens when you multiply the same bases?
3⁴ ⋅ 3² = 3⋅3⋅3⋅3⋅3⋅3 or 3 ⁶
We just add the exponents if the bases are the same!!

Well then, what about (3⁴)² ?
Wait.. look carefully isn't that saying 3⁴ Squared?
That would be (3⁴)(3⁴), right?
.. and looking at the rule above all we have to do here is then add those bases or 4 + 4 = 8 so the answer would be 3⁸.
OR
we could have made each (3⁴) = (3⋅3⋅3⋅3)
so (3⁴)² would be 3⋅3⋅3⋅3⋅3⋅3⋅3⋅3 or still 3⁸
But wait... isn't that multiplying the two powers? So when raising a power to a power-- you multiply!!
(3⁴)² = 3⁸

1 to any more is still just 1
1⁵ = 1

0 to any power is still 0!!

Evaluate if a = 3 and b = 5
Just substitute in... but use hugs () we all love our hugs!!
a³ + b²
would be (3)³ + (5) ²
= 27 + 25 = 52




Check out this great Video on the Powers of Ten
POWERS OF TEN

Algebra Honors ( Period 6 & 7)

Multiplying Polynomials by Monomials 4-5
This is just the distributive property
x(x + 3) = x² + 3x

-2x(4x² - 3x + 5)

-8x³ + 6 x² -10x
The book shows you how to multiply using a vertical method but I think using the original method taught with the distributive property works just as well-- if not better.

n(2-5n) + 5(n² -2 ) = 0
2n - 5n² + 5n² - 10 = 0
2n - 10 = 0
2n = 10
n = 5
and in set notation {5}

1/2(6xc + 4) -2(c + 5/2) = 2/3 (9-3c)
3c + 2 - 2c - 5 = 6 - 2c
3c -3 = 6
3c = 9
c = 3
and in set notation {3}

Multiplying Polynomials  4-6
This is just double distributive property or triple distributive property so you really need to understand the DP

You have learned how to use the DP to multiply
2x(3x + 2)
but... what happens if you had instead
(2x +5)(3x + 2)
There are a number of different strategies to simplify this multiplication
(2x +5)(3x) + (2x + 5)(2)

6x² + 15x + 4x + 10

6x² + 19x + 10

You could also use Fireworks—as show in class or FOIL

Fà First
OàOuter
IàInner
Là Last

 (2x + 5)(3x + 2)

The F is the first terms   (2x)(3x)

The O is the Outer terms (2x)(2)

The I is the Inner terms (5)(3x)

the L is the Last Terms (5)(2)

or 6x² + 4x + 15x + 10

6x² + 19x + 10

Example:

(3x-2)(2x²- 5x-4)

The book shows you how to multiply in vertical form, similar to how you multiply multi-digit  numbers.

Read Page 161 if you are interested in reviewing that strategy

Step 1   2x² – 5x – 4                         Step 2                       2x² – 5x – 4                   

3x – 2                                                                         3x – 2             

6x³-15x² – 12x                                                    6x³-15x² – 12x

                                                                                                 -4x² +10x + 8     

Step 3        Add:

  2x² – 5x – 4                                                                        

   3x – 2             6x³-15x² – 12x                                                    

       -4x² +10x   + 8     

6x³ – 19x² -2x + 8

I showed Fireworks and Double Distributive Property with this example as well

but also showed my favorite… The BOX Method

Create a box as big as the polynomials

In this case it’s a 2 by 3

We talked about the order of the polynomials.

Make sure to place them in descending order.

The book terms it decreasing degree of x:

We discussed

x³ -3x² + xy² + 2y³

To see the advantage of rearranging terms, multiply the polynomial

(y +2x)(x³ – 2y³ + 3xy² + x²y)

We then rearranged both polynomials into decreasing degree in terms of x

(2x + y)(x³ + x²y+ 3xy² – 2y³)

Using the BOX Method

we could find the simplified form to be

2x⁴ + 3x³y + 7x²y² –xy³ -2y⁴

Math 7 ( Period 4)

Chapter 1.5 relating to Chapter 2

There are 2 formulas in Geometry that are used with equations a great deal:

Perimeter and area

PERIMETER – the sum of all the sides of a polygon

AREA- ( we are only focusing on squares, rectangles and triangles)

Area of a square =  s² where s represents the length of the square

Area of a rectangle = bh or lw

Area of a triangle = bh/2  or ½ bh

(We discussed how we were able to arrive at the formula for a triangle…)

Compound Shapes

Sometimes a shape will be composed of several shapes. Right now we are covering shapes that are composed of squares rectangles and triangles. You break the shape into the parts you know. IF there is a side missing a measurement, you may have to SUBTRACT or ADD using the known measurements of other sides.

We looked at Page 24…

In Chapter 2 you will use the formula to find a missing side using equation solving!

If you know the area of a rectangle is 20 feet squared and one of the sides is 2 feet, what is the other side?

Since A = bh   you can solve for the missing side by plugging in the area and the known side

20 = 2h

h = 10

10 feet

An example of a triangle

The area of a triangle is 20 feet squared. The base of the triangle is 10 feet. How long is the height?

A = b/2

20 = 10h/2

20 = 5h

h = 4

The height must be 4 feet

Always remember the formula of a triangle is half the formula for a rectangle with the same base and height!

This one works nicely because the base was even and we could divide by 2 easily. You can also solve triangles with a base or height that is odd—but we will focus on that later with 2-step equations and equations with fractional coefficients.  

Algebra Honors (Periods 6 & 7)

Powers of Monomials 4-4
To find a power of a monomial that is already a power, you can use the definition of a power and the rule of exponents for products of powers.

(x⁵)³ = x⁵∙ x⁵∙ x⁵ = x⁵⁺⁵⁺⁵= x¹⁵

Notice that  (x⁵)³ = x¹⁵ or x^5∙3

In general

(a^m)ⁿ = a^mn

Rule of Exponents for a Power of a Power

For all positive integers m and n

(a^m)ⁿ = a^mn

To find a power of a power, you multiply the exponents

(u⁴)⁵ = u²⁰

[(-a)²]³ = (a²)³= a⁶

To find the power of a product, you can use the definition of a power AND the commutative and associative properties of multiplication.

(2x)3 = (2x)(2x)(2x) = (2∙2∙2)∙ (x∙ x∙ x)

= 2³∙x³

=8x³

Notice BOTH the 2 and the x are cubed

So (ab)^m = a^mb^m

Rule for Exponents for a Power of a Product

For every positive integer m

(ab)^m = a^mb^m

To find a power of a product, you find the power of each factor and then multiply

(-2k)⁵=(-2)⁵k⁵=-32k⁵

Simplify

(-3x²y⁵)³ = (-3)³(x²)³(y⁵)³

= -27x⁶y¹⁵

Negative Exponents 
If a is a nonzero real number and n is a positive integer

a^-n = 1/aⁿ

so
 10^-3 = 1/10³ = 1/1000
( remember this from Math 6 A)  

5^-4= 1/5⁴ = 1/625

16^-1 = 1/16

Let's look at the rule of exponents for division

Rule of Exponents for Division

If a is a non zero real number and m and n are positive integers, then:

If m>n

a^m/aⁿ = a^m-n 

If n > m

a^m/aⁿ = a^m-n  

but that means it would be 1/a^n-m

If m = n

a^m/aⁿ = a^m-n  = a⁰ = 1

Let's look at the 2nd case using an example

x²/x⁷ =x^2-7 = x^-5 but we write that without negative exponents as 1/x⁵

Looking at the rule above if  n > m

And in this case 7> 2

It says  that the simplified form would be 1/x^7-2 which is 1/x⁵

 This should  help you understand why
a^-n = 1/aⁿ
recall that for m > n a^m/aⁿ= a^m-n
More examples
a⁷/a³ = a^7-3= a⁴
you can also apply this rule when m < n that is when m - n becomes a negative number. For example a³/a⁷ = a^3-7 = a^-4
since
a⁷/a³ and a³/a⁷ are reciprocals then
a⁴ and a^-4 must also be reciprocals.
Thus
a^-4= 1/a⁴
a⁵/ a⁵ = a^5-5 = a⁰
But you already know that a⁵/a⁵ = 1
SO, definition of a⁰
a⁰ = 1
However, the expression 0⁰ has no meaning

All the rules for positive exponents also hold for zero and negative exponents.

Summary of Rules for Exponents 

Let m and n be any integers
Let a and b be any non zero integers
We will be reviewing these throughout the year.. but I want you to begin to understand this concept! We will be incorporating it into an upcoming Project!!

Products of Powers 

b^mbⁿ = b^m+n

Example with negative exponents
2³⋅2^-5 = 2^3+(-5) = 2^-2 = 1/2² = 1/4

Quotients of Powers 

b^m ÷ bⁿ = b^m-n

Example with negative exponents
6³÷6⁷= 6^3-7= 6^-4= 1/6⁴= 1/1296

Power of Powers

(b^m)ⁿ = b^mn

Example with negative exponents
(2³)^-2 = 2^-6 = 1/2⁶ = 1/64

Power of  a Product

(ab)^m= a^mb^m

Example with negative exponents
(3x)^-2 = 3^-2 ⋅x^-2 = 1/3²⋅1/x² = 1/9x²

Power of  a Quotient

(a/b)^m= a^m/b^m

Example with negative exponents
(3/5)^-2= 3^-2/5^-2= (1/3²)/ (1/5²)= 1/3² ÷ 1/5² which means
1/3² ⋅5²/1= 5²/3²= 25/9

Math 6A ( Periods 1 & 2)

Writing Inequalities 2-3

2 < 7 and 7 > 2 are two inequalities that state the relationship between the numbers 2 and 7

2 < 7 reads 2 is the less than 7 and
7 > 2 reads 7 is greater than 2
The symbols < and > are called inequality symbols.

Notice the mathematical sentence (inequality)
Two is less than ten or 2 < 10 is different from the mathematical phrase (expression)
Two less than ten. 10 - 2
A number 2 + x is greater than a number t 2 + x > t

The point of the number line that is paired with a number is called the graph of that number.

Check out the graph in the middle of page 39 of our textbook. When you graph numbers on the number line, make sure to place a dot DIRECTLY ON the number line at that particular number's location.
Again, check out our textbook for examples!!

Looking at the graph of numbers, we see that the larger number will be to the right of the smaller number.

A number n is between 6 and 12 so 6 < n < 12 or 12 > n > 6

Thursday's Lesson: continuing on ....

Notice the subtle differences in the sentence
Six is greater than a number t
and the phrase
six greater than a number t

Six is greater than a number t becomes 6 > t
while
six greater than a number t becomes t + 6

What about the following inequality:
A number p is greater than a number q
is p > q

The value in cents of d dimes is less than the value in cents of n nickels.

If you need to-- set up your T-charts (refer to your class notes) one for dimes and the other for nickels.

10d represents the number of dimes and 5n represents the number of nickels

so 10d < 5n

Algebra Honors ( Periods 6 & 7)

Multiplying Monomials 4-3
POWER RULES:


MULTIPLYING Powers with LIKE BASES:

Simply ADD THE POWERS

m⁵m³ = m⁸

You can check this by EXPANDING:
(mmmmm)(mmm) = m⁸



DIVIDING Powers with LIKE BASES:

Simply SUBTRACT the POWERS

m⁸/m⁵ = m³     


Again, you can check this by EXPANDING:
mmmmmmmm/mmmmm = mmm

ZERO POWERS:

Anything to the zero power = 1


(except zero to the zero power is undefined)


Proof of this was given in class:

1 = mmmmmmmm/mmmmmmmm
= m⁸/m⁸
= m⁰ (by power rules for division)
    


By the transitive property of equality : 1 = m⁰


Review 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 = -32

(-2)⁴ = +16



IF THERE IS A NEGATIVE BUT NO PARENTHESES:

ALWAYS NEGATIVE!!!!

-2⁵ = -32

-2⁴ = -16

JUST REMEMBER
NEGATIVE POWERS MEANS THE NUMBERS ARE FRACTIONS


They're in the wrong place in the fraction

m³/m⁵ = m^-2
     

m³/m⁵ = mmm/ mmmmm
= 1/mm


Again, by transitive property of equality:

m³/m⁵ = m^-2 = 1/m²

Remember the rule of powers with (  )
When there is a product inside the (  ), then everything inside is to the power!

If there are no (  ), then only the variable/number right next to the power is raised to that power.

3x^-2 does not equal (3x)^-2
The first is 3/x² and the second is 1/9x²

RESTATE A FRACTION INTO A NEGATIVE POWER:

1) Restate the denominator into a power

2) Move to the numerator by turning the power negative


EXAMPLE: 
1/32
 = 1/(2)⁵ = (2)^-5

Math 7 ( Period 4)

Solving Equations Using All for Ops 2.5 and 2.6

What’s the GOAL?  Determine the value of the variable?

How?  Isolate the variable—get it alone on one side of the equation

What do I do? Use inverse (opposite operations to “get rid” of everything on the side with the variable

What should my focus be?  When equations get really complicated…. ALWAYS focus on the variable FIRST!

One Step Equations with all 4 Operations

 –We will be meeting 4 new BFF’s

Equation Balancing Properties of Equality – there are 4 of them

Whatever YOU DO to BALANCE an equation – that operation is the property of equality that was used.

If you have…

X + 3 = 10  …you used the subtraction property  of Equality because you need to SUBTRACT 3 from both sides equally.

x – 3 = 10 …you used the Addition Property of Equality because you need to ADD 3 to each side equally

3x = 9 … you used the Division Property of Equality because you need to DIVIDE  each side by 3 equally

x/3 = 9…  you used the Multiplication Property of Equality because you need to MULTIPLY both sides by 3 equally.

TWO more Properties

Additive Inverse
using the opposite sign of the term given in the equation results in the term dropping out because it simplifies to 0

Multiplicative Inverse
using the reciprocal of the term given in the equation results in the term dropping out because it simplifies to 1

Here are the steps and justifications

1. Focus on the side where the variable is and focus specifically on what is in the way of the variable being by itself (isolated)

2. What is the operation that the variable is doing with that number in its way?

3. Get rid of that number by using the opposite (inverse) operation

Golden Rule of Equations:

Whatever you do to one side do unto the other side

Doing the same thing on both sides is actually the new set of properties (your new BFF’s)

When you multiply equally, it’s the multiplication property of equality

When you divide equally, it’s the division property of equality

When you add equally, it’s the addition property of equality

When you subtract equally, it’s the subtraction property of equality

4. JUSTIFICATION  You have just used one of the properties of equality… Which one?

Whatever operation you used to balance both sides (not the operation of the original equation)

5.You should now have the variable all alone (isolated) on one side of the equal sign.

6. JUSTIFICATION?  Why is the variable alone?  For + and -, you used the Identify Property of Addition ( ID+) which simply means that you don’t bring down the ZERO because when you add ZERO it does not change anything! ( NOTE:  There is no ID of subtraction)

For ×   and  ÷  you used the Identity Property of Multiplication ( ID x) which simply means that you don’t bring down the ONE because when you multiply by one it doesn’t change anything ( NOTE: There is no ID of division!)

FORMAL CHECK:

There are three steps to a formal check:

1. Rewrite the original equation FROM THE ORIGIANL SOUSE – this is just o case that ou find out you copied the problem wrong

2. Substitute your answer where the variable is and QUESTIO you answer by placing a “?” over the equal sign

3. REALLY  Do the math and finally set both sides equal. Place a check mark, a happy face… to indicate that you really did check this!

Math 7 ( Period 4)

Solving Equations With Mental Math 2.3

An identity equation is where ANY number can be substituted for the variable.
The equation will be TRUE.

What will happen is that while you are balancing the equation, you will ultimately end up with the same exact expression on each side. The Distributive Property is a good example of an IDENTITY EQUATION

3(x + 4) = 3x + 12

Since DP ALWAYS works that means that ANY NUMBER will for x

A conditional equation is one that has a variable where only certain values will make the equation true.

3x = 12 is conditional because 3x = 12 if and only if x = 4

You can use mental math to guess and check to solve simple equations 5y = 35

Instead of solving you think “what number times 5 will give me 35?

y = 7

Now you plug in your GUESS and CHECK that you are right> 5(7) = 35

Yes  7 is the solution

Algebra Honors ( Periods 6 & 7)

Adding  & Subtracting Polynomials 4-2

Terms to Know:
Polynomials = SUM of monomials

Monomials must have variables with whole number powers.
no variables in the denominator, no roots of numbers!!
so 1/x is not a monomial
neither is x ^1/2

constants have whole number power of zero..
7 is really 7x⁰
constants are also called constant monomials


1 term = monomial
2 terms = binomial
3 terms = trinomial

TERMS are separated by addition
( if see subtraction-- THINK: add the opposite!!)

Coefficient - number attached to the variable ( it can be a fraction)
3x² - 10x
the coefficients are 3 and -10. Make sure to attach the negative sign to the coefficient -- and ADD the OPPOSITE

y/6 is really (1/6)y so the coefficient is 1/6
if you have -x/3 that is really (-1/3)x so the coefficient is -1/3


Constant = the number that is not attached to ANY variable

Two monomials that are exactly alike ( except for the coefficient) are said to be similar, or like, terms.
-5xy² and xy² and (1/3) xy² are like terms. So is 16yxy because when you combine it 16yxy becomes 16xy²
But... -3xy² and -3x²y are NOT!!!

A polynomial is simplified when no two terms are similar
-6x³ + 3 x² + x² + 6x³ - 5 can be simplified to
4x² - 5



Some MORE TERMS YOU NEED TO KNOW
Degree of a variable in a term = number of times that variable occurs as a factor.
Degree of a term = SUM of the exponents of all its variables
-6x⁴ : the degree is 4
8x² : the degree is 2
-2x : the degree is 1
9 : the degree is 0 ( think 9 is really 9x⁰

Degree of a polynomial - HIGHEST degree of any of its terms
so
-6x⁴ + 8x² + -2x + 9
The degree of the polynomial is : 4

Leading term = term with the HIGHEST degree
Leading coefficient- the coefficient of the leading term

Day Two of this lesson:

Descending order- write the variables with the highest power first ( This is the way it is usually written)

Ascending order- write the variables with the lowest power first ( actually NEVER used in practice)


Adding Polynomials
This is nothing more than combining LIKE TERMS
LIKE TERMS = same variable AND same power

You can either do this using 3 different strategies:
1. Simply do it in your head, but keep track by crossing out the terms as you use them.
2. Rewrite putting the like terms together (commutative and associative property)
3. Rewrite in COLUMN form, putting like terms on top of each other like you do when adding a column of numbers.

EXAMPLE OF COLUMN FORM:
(5x⁴ - 3x² - (-4x) + 3) + (-10x⁴ + 3x³- 3x² - x + 3)
Rewrite in column form, lining up like terms:


Subtraction of Polynomials
You can use the ADDITIVE INVERSE PROPERTY with polynomials!
Subtracting is simply adding the opposite so.............
DISTRIBUTE THE NEGATIVE SIGN TO EACH TERM!!
(Change all the signs of the second polynomial!)
After you change all the signs, use one of your ADDING POLYNOMIAL strategies!
(see the 3 strategies listed above under Chapter 5-7)

EXAMPLE OF COLUMN FORM:
(5x⁴ - 3x² - (-4x) + 3) - (-10x⁴ + 3x³- 3x² - x + 3)
Rewrite in column form, lining up like terms:
5x⁴ - 3x² - (-4x) + 3
- ( -10x⁴ + 3x³- 3x² - x + 3)
-----------------------------------

For the sake of showing you here, I have added ZERO Terms to line up columns
+ 5x⁴ + 0x³ - 3x² -(-4x) + 3
-(-10x⁴ +3x³- 3x² - x + 3)
-----------------------------------

DISTRIBUTE THE NEGATIVE, THEN ADD:
5x⁴ + 0x³ - 3x² - (-4x) + 3
+10x⁴ -3x³ +3x² + x - 3
-----------------------------------
15x⁴ - 3x³ + 5 x

Algebra Honors ( Periods 6 & 7)

Exponents 4-1
In the expression 5⁴ the number 4 is called the exponent and the number 5 is called the base
We call 5⁴ the exponential form of 5⋅5⋅5⋅5 (which is the expanded form)
The exponent tells you the number of times the base is used as a factor.
bⁿ = b⋅b⋅b⋅b⋅b⋅⋅⋅⋅b ( a total of n factors)
The expression bⁿ tells you that b is used as a factor n times.

-2⋅p⋅q⋅3⋅p⋅q⋅p written in exponential form is -6p³q²

Be careful when an expression contains both parentheses and exponents
(2y)³ = (2y)(2y)(2y) = 8y³
and
2y³ means 2⋅y⋅y⋅y = 2y³
-3⁴ = -81
(-3)⁴ = (-3)(-3)(-3)(-3) = +81
(1 + 5)² = 36
1 + 5²= 26
Simplify
(x –y)³/2x + y
for x = 2 and y = 5

-3

Simplify
x⁴ – a⁴
for x = 2 and a = 3
-65

[2³ + 3³] ÷ [2³ + (-1)²] = (8 + 27) /9 = 35/9


Evaluating terms - this is what we have been doing all year... plug it in, plug it in!!
Remember to ALWAYS put the number you substitute in parentheses!!

2x²y + 5xy - 4, where x = -4 and y = 5
Substitute carefully:

2(-4)²(5) + 5(-4)(5) - 4
= 2(16)(5) +(-20)(5) - 4
= 160 +(-100) -4
= 160 -104
= 56

Math 6A ( Periods 1 & 2)

Writing Mathematical Equations 2-2

The process of writing equations really is just writing two equal expressions and joining them by an equals sign. The words "is" "equals" or "equal to" all indicate that two phrases NAME the same number.
The equals sign is the VERB in a mathematical sentence-- without it you have a mathematical expression!!
Eight increased by a number x is equal to thirty-seven.
In translating this mathematical sentence, I always start but placing the equals sign directly under the words "is equal"
so my first step would be

Eight increased by a number x is equal to thirty-seven.
----------------------------------- = ------------------
Then I would translate each mathematical phrase separately.
Yes, thirty-seven is a mathematical phrase!!
8 + x = 37

Ten is two less than a number n
10 = n - 2

Twice a number w equals the sum of the number and four
2w = w + 4

Notice that when you are indicating multiplication the number (coefficient) ALWAYS is placed in front of the variable.

So three times a number b would be 3b
The only time you see the letter first-- is when you are looking for our ROOM--
which is K 101... a mathematician did not label the room numbers!! :)

Sometimes we need to write an equation for a word sentence that involves measurements. MAKE SURE that each side of the equation uses the SAME UNIT of Measurement!!
For example. Write an equation for: The value of d dimes is $27.50
WE know that the value of d dimes is 10d cents (from our previous lesson) .. but $27.50 is in terms of dollars so we need to change it to cents . $27.50 is 2750 cents ... So our equation becomes 10d = 2750.