Thursday, August 30, 2012

Remember when calculating the area of a polygon or greater than 4 sides, it's easiest to use the formula,

A = (1/2)ap, where "p" is the perimeter and "a" is the apothem.

Recall that perimeter is the total distance of the sides of the polygon.  The apothem is the distance from the center of the polygon, perpendicular to the base. 

Tuesday, August 28, 2012

When we evaluate binomial expressions raised to a power, there is a pattern that develops. For example, we take the
binomial expression
(x + y) and raise it to the nth power, where n = 1, 2, 3, …...

(x + y)1 = x + y

(x + y)^2 = (x + y)(x + y) = x^2 + xy + xy + y^2 = x^2 + 2xy + y2

(x + y)^3 = (x + y)^2(x + y) = (x^2 + 2xy + y^2)(x + y) = x^3 + x2y + 2x^2y + 2xy^2 + y^2x + y^3 = x^3 + 3x^2y + 3xy^2 + y3

(x + y)^4 = (x + y)^3(x + y) = (x^3 + 3x^2y + 3xy^2 + y^3)(x + y) = x^4 + x^3y + 3x^3y + 3x^2y^2 + 3x^2y^2 + 3xy^3 + y^3x + y4

=x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y4

Notice that the expansion of each of the binomials is a polynomial. Some patterns observed are as follows:
1. Each expansion begins with the term
2. The exponents on x decrease by 1 with each term.

3. The exponents on y begin with 0 (Since y0 = 1 there is no y in any first term)
4. The exponents on y increase by 1 with each term.
5. The sum of the exponents on any term always equals n.
6. The number of terms is always 1 more than n.

Let's illustrate the preceding 6 patterns with the expansion of (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y3
1. The first term is x^3. Since n = 3, it holds true that xn = x^3.

2. The exponents for x on the next terms are 2, 1 and 0, which holds true that the exponents for x decrease by 1 each term.
3. There is no y in the first term, therefore it holds true that the exponents for y begin with 0 since y0 = 1.
4. The exponents for y in the next terms are 1, 2 and 3, which holds true that the exponents for y increase by 1 each term.
5. The sum of the exponents per term are as follows: 3 + 0 = 3, 2 + 1 = 3, 1 + 2 = 3 and 0 + 3 = 3. Since
n = 3, it holds true that the sum of the exponents of any term equals n. 
6. There are 4 terms in this expansion, therefore it holds true that the number of terms is 1 more than n.

Saturday, August 25, 2012

To determine the number of positive and negative real roots, we use a technique founded by Rene Descartes, thus named
Decartes' Rule of Signs.

To find the number of positive real roots, start with the sign of the coefficient of the term with the highest power of the
variable. Count the sign changes as you proceed through the polynomial. The number of sign changes is the number of
positive real roots or less than it by a multiple of two. For example, if there are 2 sign changes there are either 2 or (2 – 2) =
0 positive real roots.

Find the number of positive real roots of f(x) = -4x^5 – 11x^4 + 2x^3 + 9x^2 - x + 3.

Starting with -4
x^5 , there is a sign change at 2x^3, another at -x and a third at 3. Therefore there are 3 sign changes and either 3 or (3 – 2) = 1 positive real roots.

To find the number of negative real roots of f(x), substitute -x into f(x) to get f(-x). Then proceed in the same manner as you would when finding the number of positive real roots. In the above example,

f(-x) = -4(-x)^5 – 11(-x)^4 + 2(-x)^3 + 9(-x)^2 - (-x) + 3

f(-x) = 4x^5 - 11x^4 – 2x^3 + 9x^2 + x + 3

Starting with 4x^5 , there is a sign change at 11x^4 and another at 9x^2 . Therefore, there are 2 or (2 – 2) = 0 negative real roots.

• Note that the number of roots in a polynomial function equals the highest power. In the above example, there are 5
roots. The sum of the positive and negative roots will not always equal the highest power. In such cases, those
polynomial functions have imaginary roots

Friday, August 24, 2012

Most of us know the sum of the degrees of the interior angles of a triangle and quadradrilateral are. But how do we determine the sum of the degrees of the interior angles of a pentagon, hexagon, ocotogan, etc? 

If n is the number of sides of the polygon, then (n - 2)180 is the sum of the interior angles of that polygon.

Therefore, the sum of the interior angles of a pentagon (5 sides) is (5 - 2)180 = 540

The sum of the interior angles of a hexagon (6 sides) is (6 - 2) 180 = 720

For a 7 sided polygon (7 - 2)180 = 900

and so on.

If the formula is hard to remember, you notice that as the number of sides increase, the sum increases by 180.

You may also want to note that the sum of the measures of the exterior angles of a polygon is always 360.

Monday, August 20, 2012

Don't get theoretical probability and experimental probability confused. 

Theoretical probability is the probability you'd expect from the outcome of an event based on theoretical principles.

For example, if you roll a fair 6 sided die, theoretically each side is equally likely to land face up. So the theoretical probability of obtaining a 1, 2, 3, 4, 5 or 6 is 1/6.

The experimental probability would occur when actually rolling the die a certain number of times and seeing how many times each side landed face up.  If you roll the die 10 times and a 1 lands face up 2 times, the experimental probability is 2/10, or 1/5. This is slightly higher than the theoretical probabiliy of 1/6.

Friday, August 17, 2012

Suppose you have triangle ABC with the given information.

The measure of angle A = 65
Meansure of angle B = 35
Length of side a = 12

What is the measure of angle C and the lengths of sides b and c?

If this was a right triangle, we could simply use sine, cosine of tangent to get side b or c and the other side would be obtained using the Pythagorean Theorem.

But since this is not a right triangle (measure of angle C = 80), we can use the Law Of Sines which states

SinA/a = SinB/b = SinC/c (alternatively it be can be written a/SinA = b/SinB = c/SinC)

We will use

SinA/a = SinB/b to get the length of side b


Sin(65)/12 = Sin(35)/b

12*Sin(35) = b*Sin(65)

6.88 = 0.906b

b = 7.6

Now we can get c using

SinA/a = SinC/c

Sin(65)/12 = Sin(80)/c

12*Sin(80) c*Sin(65)

11.82 = 0.906c

c = 13.05

Wednesday, August 15, 2012

Here is an article I wrote a few years ago on Hope it helps those having difficulties with square roots.

How To Calculate Square Root

The first method I use when teaching students how to calculate square root is to look to see if the number is a perfect square first. I suggest students memorize the perfect squares from 1 to 25: 1, 4, 9, 16, 25, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625. So if you are asked to calculate the square root of 441, you automatically know the answer is 21, or -21, since a negative number times another negative number yields a positive number.

For non perfect squares or larger numbers which you are unsure of if the number is a perfect square, I suggest using a factor tree. For example, suppose you want to know what the square root of 48 is. It is not a perfect square since 6 times 6 equals 36 and 7 times 7 equals 49. No whole number times itself equals 48. So break it down into factors. I always suggest trying to find a perfect square as one of the factors and in this case, 16 times 3 equals 48 and 16 is a perfect square. Remember since you are dealing with square root that the factors are also square root. So the square root of 48 equals the square root of 16 times the square root of 3. Three is a prime number so you cannot break down 3 any farther using a factor tree. We know the square root of 16 is 4, so the answer is 4 times the square root of 3.

Another more difficult example, say we need to get the square root of 2025. With such a large number, most people won't know if this is a perfect square, so use the factor tree. Know that any number ending in 5 is divisible by 5. So 5 times 405 is 2025. But 405 can be broken down into factors, using the same rule, 5 times 81 is 405. So far then we have the square root of 5 times the square root of 5 times the square root of 81. Notice then that square root of 5 times the square root of 5 equals the square root of 25. Now this problem becomes simple because you notice we have 2 perfect squares here, 25 and 81. Square root of 25 is 5 and square root of 81 is 9. So the answer is 5 times 9, which is 45 and -45, since -45 times -45 equals 2025. This problem is actually a perfect square, but if you do not recognize it as such, you can use the factor tree method I just described.

When dealing with the square root of a negative number, imaginary numbers come into play. The square root of -1 equals imaginary number denoted as "i". So in the above problems if we had the square root of -48, you have square root of -1 times square root of 3 times square root of 16. The answer is 4i times the square root of 3, and -4i times square root of 3. If we have the square root of -2025, the answer is simply 45i and -45i

Hope my method helps you when trying to figure out the square root of both positive and negative numbers.

Monday, August 13, 2012

Here's a few word problems from a sample GED exam to consider.

A basketball team has won 50 games of 75 played. The team still has 45 games to play. How many of the remaining games must the team win in order to win 60% of all games played during the season?

First, get the total number of games the team plays, which is 75 + 45 = 120. Now you know that the team won 60% of its games. So take 120 x 0.60 = 72, this is the total number of games the team wins. Since the team already has 50 wins, it wins 22 more games to get to 72 total wins.

Note that we use 0.60 for 60% because to change a percent to a decimal, move the decimal point 2 places to the left. You can also change 60% to 0.60 by taking 60/100.

A rectangle and a triangle have equal areas. The length of the rectangle is 12 inches, and its width is 8 inches. If the base of the triangle is 32 inches, what is the length, in inches, of the altitude drawn to the base?

Area of rectangle = length x width
Area of triangle = 1/2(base)x(height)

Area of rectangle = 12 x 8 = 96
Area of triangle = 1/2(base) x (height)
96 = 1/2(32) x height
96= 16 x height

96/16 = height

6 = height

Saturday, August 11, 2012

I wonder why the United States hasn't adopted the use of the Metric System.

After all, everything is based off of powers of 10.

1 cm = 10 mm
1 m = 100 cm
1 km = 1000 m


What do we have?

12 inches = 1 ft
3 feet = 1 yard
5280 feet = 1 mile

Wednesday, August 8, 2012

To illustrate the concept of similar triangles and the use of proportions, I found a great example from a sample GED test online.

To measure the distance (DC) across a pond, a surveyor takes points A and B so that AB is parallel to DC. If AB = 60 feet, EB = 48 feel, and ED = 80 feet, find DC. 72 ft., 84 ft. ,96 ft., 100 ft., Not enough information is given.

The important thing is that since AB and DC are parallel angles ABE and BDC are congruent (alternate interior angles) as are angles ACD and BAC congruent by the same reason. You will quickly notice that triangles ABE and DEC are similar triangles.  Because of this similarity, corresponding sides are in proportion. DE matches up with  BE, and DC matches up with AB... Create proportions with these sides and set them equal to each other.


80/48 = x/60

To solve this we can multiply the equation by 60

(80/48)(60) = (x/60)60

We multiply by 60 to get x by itself

4800/48 = x

100 = x

Saturday, August 4, 2012

Remember when the result of an equation is a true statement, the solution is "all real numbers". It the result of the equation is a false statement, the solution is "empty set", "null set" or "no solution".

For example,

3x + 4 = -2x + 5x + 4

3x + 4 = 3x + 4

This is a true statement, so the solution for x is all real numbers.

In the equation -2x + x - 6 = 6x - 4 - 5x

we get

x - 6 = x - 4

subtract an x from both sides and we have

-6 = 4.

It's obvious that it is a false statement, therefore there is no solution.