Claw Education Services

Preparing for North Carolina Math-3

Polynomial Division

This is when you divide one algebraic expression by another. It can be helpful/used-by computer programmers to know if there will be a situation when there will be a division by zero error.

There are two methods of performing this task. The traditional method works every time under every circumstance. Synthetic division only works if you are dividing by a linear expression (a variable to the first power with a constant).

Traditional Method

1 - Arrange your term/numerator from the greatest exponent to the constant. If there is a missing exponent, add it with a coefficient of zero.

2 - Do the same thing with the divisor/denominator.

3 - Looking at the first term in the numerator and the first term in the denominator, find the term by which the denominator's term must be multiplied to equal the numerator's term.

4 - Multiply the entire denominator by this term, and subtract it from the numerator.

5 - Use the remaining difference as the new numerator, and repeat steps 3 - 5 until the difference is zero or the term you added to the quotient is a constant. If the latter is true, you have a remainder. Put the remainder atop the divisor as the final part of your quotient.

Example

2x^3 - 3x^2 + 4x -5
----------------------
x^2 + 2

As the first term of the numerator is 2x^3, and the first term of the divisor is x^2, you want to multiply the divisor by 2x.

2x^3 - 3x^2 + 4x - 5
-(2x^3 + 4x)
--------------------------
-3x^2 - 5

Since the first term of the difference is -3x^2, and the first term of the divisor is x^2, you multiply the divisor by -3.

-3x^2 - 5
-(-3x^2 - 6)
---------------
1

Your remainder is 1/(x^2 + 2), and your complete quotient is:

2x - 3 + 1/(x^2 + 2)

Another Example

2x^3 + 3x^2 - 3x - 2
-----------------------
2x + 1

With the first term of the numerator being 2x^3, and the first term of the denominator being 2x, the denominator has to be multiplied by x^2.

2x^3 + 3x^2 - 3x - 2
-(2x^3 + 1x^2)
-------------------------
2x^2 - 3x - 2

With the first term of the numerator being 2x^2, and the first term of the denominator being 2x, the denominator has to be multiplied by x.

2x^2 - 3x - 2
-(2x^2 + 1x)
-----------------
- 4x - 2

With the first term of the numerator being -4x, and the first term of the denominator being 2x, the denominator has to be multiplied by -2.

-4x - 2
-(-4x - 2)
----------
0

There is no remainder, so the quotient is x^2 + x - 2

Synthetic Division

This method only works when dividing by a linear expression. If you have a polynomial in the denominator that can be factored into multiple linear expressions (binomial terms), you can user this method using the quotient and the next binomial.

1 - Arrange your term/numerator from the greatest exponent to the constant. If there is a missing exponent, add it with a coefficient of zero.

2 - Set the linear expression by which you are dividing to zero, and solve for the variable. This becomes your "multiplier."

3 - List the coefficients from your numerator.

4 - Bring-down the left-most coefficient.

5 - Multiply this coefficient by the multiplier, and add it to the next coefficient.

6 - Bring-down the sum, and repeat step 5 until you have added a product to the constant from the original term.

7 - If the final sum is not zero, you have a remainder.

8 - If there was a coefficient in your divisor, divide all terms that you just found by that coefficient.

9 - Decrease the highest exponent from the original numerator by one, and connect these variables to the numbers you just found.

Example

2x^3 + 3x^2 - 3x -2
----------------------
2x + 1

Determine your "multiplier" by setting 2x + 1 = 0. Solving for x, you have x = -1/2.

Your coefficients are:

2 3 -3 -2

Bring-down the 2, and multiply it by -1/2 to get -1.

3 + (-1) = 2

Multiply 2 by -1/2 to get -1.

-3 + (-1) = -4

Multiply -4 by -1/2 to get +2.

-2 + (+2) = 0

Your original numerator had the greatest variable ("degree") of 3; therefore, your first term of your quotient has an x^2.

2x^2 + 2x - 4 with no remainder is NOT your answer, because the denominator had a coefficient of 2; therefore, this answer has to be divided by 2 to get:

x^2 + x - 2

This is the same quotient as the second example using the traditional method.

*** NOTE ***
Step 8 of this method often is skipped by teachers. Don't make this mistake!!

Preparing for North Carolina Math-3

Exponential Growth with Financial Literacy

Exponential Growth is when items increase or decrease at a constant rate without additions or subtractions, and the increased/decreased amounts also do so. The following are two examples:

A - Exponential Growth
You put $1200 in a bank account, and earn 0.01% interest per year. This means you earn one cent interest for the first month ($1200)(.0001)(1/12) = $0.01

The next month, you earn another cent, plus a small fraction of a cent due to the interest in the account from the prior month(s).

B - Exponential Decrease
A county has a population of a million people. A pandemic strikes, and 0.01% of the population dies. This means there are 100 deaths. (1,000,000)(0.0001) = 100.

The next month, another 0.01% of the population dies, but there now are 999,900 people in the population.

F-Pint

I call the equation for Exponential Growth F-Pint.

F = Future value. This can be a higher or lower number depending if you are doing growth or decrease.

P = Present value. This is the amount of an item that you currently have.

i = Interest or rate. This is a decimal value representing the rate at which an item is increasing or decreasing. If the rate is decreasing, the value is negative.

n = Number of compounding periods. This is the number of times a growth/decrease takes place at the given rate within the rate's period of time. For example, if an annual interest rate is given, the compounding periods are:

Annual = 1
Semi-Annual = 2
Quarterly = 4
Monthly = 12
Weekly = 52
Daily = 365
Hourly = 8760

T = Number of rate's time periods. For the annual interest rate, it would be the number of years.

These fit into the equation:

F = P(1 + i/n)^(nT)

Examples

For Example A with $1200 growing at an annual rate of 0.01% compounded monthly for 20 years,

F = (1200)(1 + 0.0001/12)^(12*20)
F = (1200)(1 + 0.0001/12)^240
F = $1202.40

Note - compounding had no impact on this because the interest rate was so low and the present value was low.

For Example B with a population of a million people, and a pandemic killing 0.01% each month for three years:

F = (1,000,000)(1 - 0.0001/1)^(12 * 3)
F = (1.000,000)(0.9999)^36
F = 996,406 people remaining alive

Now is the time to start preparing your home for the summer. The following are some concepts that should save you money while keeping you cool:

Having fans pull cool air to the ceiling;

Turning blinds so they direct hot sunlight to the ceiling;

Checking doors and windows for hot air flowing into the home;

Using a dehumidifier to make indoor areas feel cool;

Using a programmable thermostat so the air compressor won't activate until an unoccupied space hits 85, and have it cool to 78 (no lower than 74) before someone is scheduled to return; and,

Using small-area coolers (portable AC units) for frequently used areas (think bedrooms) rather than cooling the entire home.

Last weekend, a banker told me that many people ask him how inflation is decreasing while they still are paying more and more for groceries and other items.

If others tell you that, remind them of eighth grade math. Inflation is the rate of INcrease. Lowering inflation just means that prices aren't increasing as quickly.

Prices stop increasing when the rate of inflation is 0%.

Today I spoke with two college students who are taking physics and calculus-2 (integral calculus). I asked them what the equation is for the volume of a sphere.

Volume = (4/3)(Pi)r^3

I then asked them what the first derivative is with respect to the radius.

(3)*(4/3)(Pi)r^2 = 4(Pi)r^2

I asked them how this is related to a sphere. It took a while, but they realized it was the equation of the area of four circles.

I then asked them what the first derivative of this equation with respect to the radius is.

(2)*(4)(Pi)r = 8(Pi)r

This is the circumference for four circles.

In the South, I was introduced to cobbler - a rectangular version of a pie.

My Geometry teacher, Frank M. Chubb, Jr., said the area of a circle is pie are square, but we knew pies are round.

Happy PI Day!!

Pi = 3.14

A quadratic curve has its minimum point at (1, 0) and a y-intercept at (0, 3). Another quadratic curve has its solutions at (1, 0) and (1/2, 0) with its y-intercept at (0, 1).

Find the two points where these curves intersect.

Jane travels three hundred miles in six hours. She then travels 715 miles at the same speed. If she left at 8:12 AM for her 715 mile trip, at what time did she reach her destination?

A - 3:27 PM

B - 10:30 PM

C - 8:06 PM

This is a multi-part problem:

The profit from the production and sale of specialty golf hats is given by the function P(x) = 20x - 8000 where x is the number of hats produced and sold.

A) Producing and selling how many hats will give a profit of $6000?

B) How many hats must be produced and sold to avoid a loss?

The average monthly bill for wireless telephone subscribers from 1985 to 2008 can be modeled by B(x) = -1.775x + 99.344, where x is the number of years since 1980. If this model remains valid, in what year will the average monthly bill be $31.89?

The future value of a simple interest investment is given by the equation S = P(1+iT). What principal (P) must be invested for four years (T) at the annual interest rate (i) of 12% so that the future value grows to $5,180?

Should students be taught to use technology before they understand the math concepts?

Once an administrator agreed with a student that I should teach them how to use the calculator before they understood the concepts. When we were in class, I gave the students a quiz using their calculators:

What is negative one squared?

What is the square root of four?

The students who used their calculators gave me the answers:

-1
2

The students who understood the concepts gave me the correct answers. What are the correct answers to these questions?

There are many environmental terms being used. Some of these are "green," "clean," and "sustainable."

Do you know the difference in these terms?

"Sustainable" means the something will last regardless of external conditions, or it naturally can be replaced. Items that burn wood are "sustainable" because wood naturally grows, and the ash is not polluting; however, burning wood creates many combustion products that are polluting.

"Green" is what you get when you mix blue and yellow. It can mean whatever the person who hears it thinks it means.

"Clean" generally means non-polluting. This means that it does not create pollution while working, and it does not generate polluting waste. "Pollution" includes products of combustion, waste, and even water v***r; therefore, "clean energy" would include hydro-electric, solar, tidal, and wind.

Are you "fiscally responsible"?

Before you can answer that question, you have to know what the words mean.

FISCALLY is a form of the word FISCAL, which means financial or money.

RESPONSIBLE means you do the right thing.

This means that "fiscally responsible" means you "do the right thing with money."

In most cases, this means that you save as much money as you can, and you don't spend more money than you have. There are times, however, when you have to spend more money than you have, which means that you owe money to someone else (have "debt").

When you go into debt, that means that the money you owe grows with interest. As you pay-off the debt, you have to pay the principal (original debt) plus all interest.

Being fiscally responsible means that you do everything you can to pay-off your debt before buying items that you may "want," but don't "need."

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www.ClawEducationServices.com

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What is Financial Literacy?

Financial literacy is a practical application of math concepts.

The most basic concept is "Balancing an Account (formerly known as balancing a check book)"

You start with an amount of money. Every time you want to buy something, or need to pay for something, you subtract that from the amount of money you have. When you earn money, you add that to the amount of money you have.

As you approach having a zero amount, you want to stop spending money until you earn more. Debit cards will not allow you to spend more money than you have. Credit cards will allow you to spend more money than you have.

Spending more money than you have is called going into DEBT. When you go into debt, the money that you owe grows using an exponential growth equation, and fees are added. This will be addressed in another post.