Monday, April 9th 2012, we met again with Mr.Marsigit, our lecture of english. He showed us again about mathematical content in video. But, i always think his videos are interesting, because we could learn mathematics with another way. Then, there is the reflection from this videos.
First, English_Greatest_Common_Factor.flv
Todays lesson want discuss about common factors and grouping.
The objectives are :
>Find the greatest common factor (GCF) of numbers
>Find the GCF of terms
>Factor out the GCF
>Factor a four_term expression by grouping
So let`s get started with some definition in particular Product and Factor, that say we have an expression :
Say that 15 is the product and the factors are 3 and 5
What we mean by factoring is that comfactoring completely means to have all factors as prime numbers.
In this case we have 15 broken down into factors 3 and 5.
Another examples is 20 = 2.2.5 ( twenty would be broken into the prime numbers of two times two times five), this number cannot be broken down any feature.
To find the greatest Common Factor (GCF) of Numbers, we are provided with a list of integers. In the list the largest common factor of the integers is noun is greatest common factors. So let`s taken an example :
45= 3.3.5= 32.5 ( fourty five can be broken down of three times three times five or three square times five)
Another number is
60= 22.3.5 ( sixty which can be broken down as two square times three times five)
We know that 32.5=45 (hree square times five is equal to fourty five). Three square is nine, nine times five is fourty five.
two square is four, four times three is twelve, twelve times five is sixty.
To find the Greatest Common Factor, choose those prime number with the smallest exponents and find their product, so GCF here is three and five.
We can say 3.5 = 15 ( three times five is fifteen) therefore 15 is GCF between 45 and 60.
Once again to find the greatest common factor (GCF) of numbers, we find the smallest exponents asosiate with the prime factors and then find their product.
So let`s consider 36, 60, 108 (thirty six, sixty and one hundred eight)
36=22..32 ( thirty six can be broken down as two square times three square). We know that two square is four and three square is nine. Four times nine is thirty six
Like way we have 60=22.3.5 (sixty can be broken down as two square times three times five). Two square is four. Four times three is twelve. Twelve times five is sixty.
And finally we have the number 108=22..33 (one hundred eight which we can be broke down as two square times three cube). Two square is four, three cube is twenty seven. Four times twenty seven is one hundred eight.
We see here two is common among of these numbers, but the exponent is two. So two square is one of the prime factors asosiate with these.
Next, we have three, three is common to all of these number however the smallest exponent is one.
Therefor, we have two square times three. So twelve is the greatest common factors among thirty six, sixty and one hundred eight.
Another way, we can take :
2 36 60 108
2 18 30 54
3 9 15 27
3 5 9 ( we can see that these cannot can be broken down)
So we have here is collect this term here, 2, 2, 3. We get the same result using this approach.
Second, English_Pre_Calculus_graph.flv
For example, f(x)=(x+2)/(x-1), then when x=1, the results is 3/0, as we know all numbers when divided by zero equal off limits.
If we draw a graph, then x=1 will break in function graph. So, don`t use x=1.
If x=0 ; f(x)=(0+2)/(0-1)=2/(-1)=-2
Now, we can draw the graph with substitute the value of x,by identify the value of f(x)
f(-3) = ¼
f(-2) = 0
f(-1) = -1/2
f(0) = -2
f(1) = ∞
f(2) = 4
f(3) = 5/2
The graph is:
Given function y = (x2 – x – 6)/(x – 3)
If x = 3, the result is 0/0. It called missing point syndrome and it’s not allowed. So, we must simplify it first.
y = (x – 3)(x + 2)/(x – 3)
y = x + 2
Third, English Math Tutor.flv
1. The Reflexive Property ofg Equality
A number is equal to itself. As the example, that there is A as a number that is equal to A, two is equal to two, and then three is equal to three.
2. The Symetric Property of Equality
If one value is equal to another, then that second value is the same as the first. As the example, there are A and B, that A is equal to B, so B is also equal to A. If there is 3 that is equal to x, so in mathematics, can be written as x is equal to 3.
3. The Transitive Property of Equality
If one value is equal to a second, and the second happens to be the same as the third, then we can conclude the first value must also equal the third. If there are A ,B, and C. And given that A is equal to B, and B is equal to C, then we can conclude that A ois equal to C.
4. The Subtitution Property
If one value is equal to another, then the second value can be used in place of the first in any algebraic expression dealing with the first value. If there are A and B , that A is equal to B, so B can subtituted for A, in the expression with A, and then A is also can subtituted for B in the expression of the B.
5. The Additive Property of Equality
We can add equal values the both sides of an equation without changing the validity of the equation.If there are A that is equal to B, so we can add the both sides with the same number, as the example, we can add the both sides with C, so the expression become A plus C is equal to B plus C, or C plus A is equal to C plus B. And the both sides are symetrys, and the expreesion can used to solve the algebraic equations.
6. The cancellation Law of Addition
If we have equation A plus C is equal to B plus C, and then we substracted the both sides by C, that equality is called cancellation law of Addition. As the result we get A is equal to B, in the other word, we are return to the first equation.
7. The Multiplicative Property of Equality
We can multiply equal values to both sides of an equation without changing the validity of the equation. If we have A is equal to B, we can multiplicate the botrh sides by C, and then we get A times C is equal to B times C, or we can also write C times A is equal to C times B. Those are the simetry equality.
8. The Cancellation Law of Multiplication
If we want to cancell the mutiplication of A and B above, then we can divide the both sides by C, so we will get A times C divided by C is equal to B times C divided by C, then the answer is A is equal to B. So we already have the first equation again.
9. The Zero-Factor Property
If two values that are being multiplied together equal zero, then one of the values, or both of them, must equal zero.We have A times B is equal zero,if that equation is true, so we can identify that A is may be zero, or B may be zero, or the both of them is may be zero. If A is equal zero when we multiplicated to B, the value is must be zero and if B is equal zero is also we will get the value is equal zero.
Properties of Inequality
1. The Law of Trichotomy
For any two values, only one of the following can be true about these values:
• They are equal
• The first has a smaller value than the second.
• The first has larger value than the second.
As the example, if given two numbers A and B, there are three possibility, first we can get that A is equal to B, and then A is less than B, or A is greater than B.
2. The Transitive Property of Inequality
If one value is smaller than a second, and the second is less than a third, then we can conbclude the first value is smaller than the third.If we have A is less than B, and B is less than C, so we can conclude that A is less than C.
3. Properties of Absolute value
The absolute of A is more than equal to zero, and the absolute of negative of A is equal to absolute of A. The absolute of multiplication of A and B is equal to absolute of A multiplicated by absolute of B. And also the absolute of A over B is equal to absolute of A over absolute of B with the contain that B is not equal to zero.
Properties of Numbers Closure
1. The Closure Property of Addition
When we add real numbers to other real numbers, the sum is also real. So the addition is closed operation.As the example, there are A and B which the both are real numbers, and if we add them, we will get real numbers too.
2. The Closure Property of Multiplication
When we multiply real numbers to other real numbers, the product is a real number. So the nultiplication is “closed” operation.If A and B are real number, so thew result of A times B is also real numbers.
But when we doing the substraction operation, we can not conclude that the substraction is “closed” operation. As the example, we have three which a natural number, and five is also a natural number, when we subtract three by five, the result is negative of two which not a part of set of natural numbers. So the subtraction is not “closed” operation.
3. The commutative Property of addition
It does not matter the order in which numbers are added together. A plus B is equal to B plus A.
4. The Commutative Property of Multiplication
It does not matter the order in which numbers are multiplied together. A times B is the same thing with B times A.
Associativity
1. The Associative Property of Addition
When we wish to add three or more numbers, it does not matter how are group then together for adding purposes. The parentheses can be placed as we wish. If there are A plus B plus C, we are free to operation A plus B firstly and then add by C, or we calculate B plus C as the first and then we add by A is not matter.
2. The Associative Property of Multiplication
When we wish to multiply three (or more) numbers, it does matter how we group them together for multiplication purposes. The parentheses can be p[laced as we wish. We can calculate the first and the second and the result is multiplied by the third also we cxan calculate the second and the third as the first, after that multiplied by the first, is not matter.
3. The Identity Property of Addition
There exists a special number, called the “additive identity”, when added to any other number, then that other number will still “keep its identity” and remain the same. A plus zero is equal A (that number itself).
4. The Identity Property of Multiplication
There exists a special number, called the “multiplicative identity”, when multiplied to any other number, then that other number will still “keep its identity” and remain the same. A multiplied by one is equal to A (the number itself).
Inverse
1. The inverse Property of Addition
For every real number, there exist another real number that is called is opposite, such that, when added together, you get the additive identity (the number zero). A plus negative A is equal zero.
2. The inverse Property of Multiplication
For every real number,except zero, there exist another real number that is called is multiplicative inverse, or reciprocal,, such that, when multiplied together, you get the multiplicative identity (the number one). A times 1 over A is equal one.
3. The Distributive Law of Multiplication Over Addition
Mutliplying a number by a sum of numbers is the same as multuiplying each number in the sum individually, then adding up our products. As the example, 5(7+3)= 5 (10)= 50, in the other way, we calculate each contain, that is 5(7) + 5(3)= 35 +15= 50.in general form , we get that A (B+C) = A(B)+A(C) and if the form of equation is (A+B)C=A(C)+B(C).
4. The Distribitive Law of Multiplication over subtraction
A(B-C)=A(B)-A(C).
The genaral distributive Property
a (b1+b2+b3+...+bn)
= ab1+ab2+ab3+...+abn
5. The Negation Distributive Property
If we negate (or find the oppsite) of a sum, just “change the signs” of whatever is inside the parentheses.
-(A+B)=(-A)+(-B)=-A-B
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