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Factorization by grouping

Until now we’ve only been factorizing polynomials in all which the coefficients share the same greatest common factor. But what happens when they don’t? Consider the following example:

2x^2 + 8x +3x +12

Just by glancing at the above mentioned example, you’d be able to see that all the four coefficients share no common factor. But if we were to look closely, we can see that the first two coefficients and the last two coefficients share a common factor. Take a look at the following:

2x^2 + 8x (Greatest common factor = 2x) AND 3x + 12 (greatest common factor = 3)

This means we can actually factor these two parts out of the polynomial. By applying the concept of grouping, we can factor the polynomial as follows:

2x(x + 4) + 3(x+4)

This in turn can be rewritten as:

(2x+3)(x+4)

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“In ancient times, having power meant having access to data. Today, having power means knowing what to ignore.” - Yuval Harari in Homo Deus

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The greatest common factor

When were talking about the greatest common factor between two monomials, we’re essentially asking what the highest factor is that is divisible in both monomials. See the illustration taken from Khanacademy to see how to solve an example:

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Factorization and divisibility

So what does the word factor mean? Let’s look at the number 12. What two numbers do we have to mulitply in order to get the number 12? Some examples are 12*1, 2*6, 4*3 et cetera. In all of these cases, we could say that 2 both and 6 in 2*6 are both factors of 12. Both 12 and 1 are factors of 12. But the numbers have to be integers.

On the contrary, we could also say that 12 is divisible by 12 and 1. In the same manner, we could also say that 12 is divisible by 3 and 4.One number is divisible by another number if the result of the division is a integer.

Now, let’s move on to the algebraic context. Firstly, let’s solve the following equation:

(x+5)(x+5)

The answer to the above mentioned equation as follows:

 x^2+10x+25

To apply the concept of factorization here, we could say the following:

x+5 is a factor of x^2+10x+25.

In parallel, we could as say the following when we apply the concept of divisibility:

x^2+10x+25 is divisible by x+5.

Not to get back to the actual name of the title, we still have discuss about how to factorize a monomial. Let’s first recall what integer factorization is:

Example: If we were to factorize 20 to its prime numbers, we would end up with: 2, 5, 2.

Integer factorization is, in other words, Decomposing an integer number into its prime numbers.

Now let’s move on to factorizing monomials:

Example: factorize 20x^3

Answer: 5*2*2*x*x*x

In the above example one could see that the monomial is decomposed to its prime factors: 20 consist of 5, 2, and 2; and x^3 consists of x*x*x.

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More about Polynomials

I’ve defined the term polynomials in a earlier post, but to refresh our minds about his topic, let’s talk about when something is not a polynomial!

What a polynomial is not

First of all, a polynomial is characterized by terms that have an exponent that is a non-negative integer. The following is, for example, not considered a polynomial because the leading term (first term in the polynomial) contains a negative exponent: 

9a^(-5)-10

The examples given below are also not considered to be polynomials because the exponents are not an integer.

9a^(1/3)-13

sqrt(9a)-13

Degree and leading terms

We will now move on to some terms that are used in the algebraic context. When asked what the highest degree of a polynomial is, it’s asking which term contains the highest exponent. For instance, in the example given below, the highest degree is 9a^5.

9a^5 + 3x + 10

As for the leading term of an expression, as written in the second paragraph, a leading term is simply the term that is positioned at the front of the expression. The leading term in the example above is 9a^5.

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The principle square root

Normally when we saw a sqrt(-1), we gave a i. Which means it is an imaginary number. Negative numbers cannot be squared, which is why we give it a imaginary number as a sort of placeholder for a number that we don’t have.

The question you’d come next to is why we can’t get the root of a negative number. Let’s see it this way, if we were to square a negative number (two times the same number) then we’d still get a positive number because two negative numbers times itself cancel each other out. And what about two positive numbers? We’d still get a positive outcome. The only way to get a negative outcome, is only when one the two numbers is negative. 

But, when we take the root of a number, the squared result of that outcome should be the same as the number from which the root is taken from. So the conclusion here is that the result of a squared root number cannot be negative. 

The next thing we’re going to look at is simplifying the square root of a negative number. If we were to have sqrt(-a), we can further simplify this by saying this: sqrt(-1*a). We can then break it down further to sqrt(-1)*sqrt(a). What we can do now is further evaluate the positive number of a in the equation. And lastly, we can finally rewrite sqrt(-1) to i. This means that we can rewrite that as i * sqrt(a) or preferably: sqrt(ai).

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The domain of a function

When the domain of a function is asked, it refers to the set of possible value inputs for which the function defined. 

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Arithmetic sequence formulas

Before we combine both topics of formulas and arithmetic sequences, let’s define what arithmetic sequences mean. As the name suggests: arithmetic sequences are sequences of arithmetic sums. An example of a positive arithmetic sum is: 2,4,6,8,10. A number in a sequence is called a term.

Now, let’s combine formulas to the equation. In addition to the way we used to describe arithmetic sequences like I did above, there is another way by utilizing formulas. The two new ways I’ll be talking about are recursive formulas and explicit formulas. Formulas give us instructions on how to find any term of a sequence, much like an array or list in programming. 

In formulas, n is used to represent any term number and a(n) to represent the n’th term of the sequence. To give an analogy from programming: n is the input for the formula and a(n) returns the outcome of the formula.

Here’s is an example taken from Khanacademy:

As we can see from the illustration above, n, representing the term number within a sequence, acts as the input for our formula while a(n) returns the result of the sequence. 

Recursive formulas of arithmetic sequences

Before we look into the what a recursive formula is, let’s look at what the recursive actually means. The word recursive, usually used in the context of mathematics and linguistics, means that something is related to or involved with repeated application of a rule. As we further explain recursive formulas, we’ll be able to see that the formula holds true to its name.

Recursive formulas provides two pieces of information:

1. The first term of a sequence;

2. The pattern rule to get any term in a sequence from the term that comes before it.

An example of a recursive formula of the following sequence: 3,5,7 is written below:

a(1) = 3 //The first term is three

a(n) = a(n-1) + 2 // Add two to the previous term which is represented by n-1.

As can be seen in the example above, we can derive 1) the first term of the formula (3); and 2) the pattern rule that is being added to the previous term (+2).

In order to find the fifth term, for example, we’d need to extend the sequence term by term. See the snapshot taken from Khanacademy below:

The only disadvantage of this is that we’d need to calculate the terms step by step in order to get to n=5 as we don’t know what the contents of the formula are.

Explicit formulas of arithmetic sequences

An example of an explicit formula is as given below:

a(n) = 3+2(n-1)

The advantage of an explicit formula as opposed to a recursive formula is that we can plug in any number of the term that we’d like to know to get the value of the term. To find the fifth term, for example, we need to plug in n=5 into the explicit formula:

As we can see from the illustration above, we are now able to read the inside-outs of the formula which we previously couldn’t with the recursive formula.  

If we were to decompose the formula, we’d see that the initial value of the formula is 3. The number that is being added every term is 2.

Sequences are functions

Since sequences are always functions, it’s not possible to input any negative or decimal numbers in the formula. This means that the domain of the sequences -- which is the set of all possible inputs of the function -- is the positive integers.

Alternative notations

Outside of the previously used a(4) notation, other sources sometimes write a4, with 4 having a smaller font. Both notations are fine to use, but Khanacademy has a preference of using a(4) because it emphasizes that sequences are functions.

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When does a graph represent a function

According to Khanacademy: a graph only represents a function if every x-value in the graph only has one y-value.

In other words, if an x-value has more than one corresponding y-value, the graph does not represent a function. (See example below taken from Khanacademy)

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Dependent and independent variables

Dependent variables are variables that are affected when you change the independent variable. Consider, for instance, the following example:

You are going for a run. For every mile you burn 100 calories. In the equation below, m represents the number of miles you run, and c represents the number of calories you burn.

c = 100m

The number of calories you burn depends on the number of miles you have run. Therefore, the number of calories you burn is the dependent variable. The number of mile you run is the indepedent variable. 

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Writing algebraic expressions

When solving algebraic problems, I’ve found that there seem to be some unusual expressions that need to be rewritten. Here’s what I found:

10 less than a 

a - 10 (substraction)

11 more than a 

a + 11 (addition)

The product of 10 and z 

 10z (multiplication)

The above examples were mere the basics. In some of the harder examples these expressions are combined in one sentence:

Four more than the product of one and a number x 

1x + 4

Six minus the product of four and a number x

6 - 4x

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不怕慢，只怕站

“有些人讀書熱情很高，但不能持久，結果收效甚微。俗話說： ‘ 不怕慢，只怕站。’ 如果你實在太忙，每天工作之餘，擠30分鍾來讀書總是可以的吧。“  - 彭子平

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Multiplying two fractions

The way to multiply two fractions with each other is to simply multiply the two numerators and denominators with each other. An example is found in the illustration below, taken from Khan Academy.

But why does that make sense? One way to see it is this: imagine having a chocolate bar that is divided in three pieces. We have two pieces that represents the numerator. Now, divide every piece of the chocolate bar in 5 pieces. Now we have 15 pieces in total. This represents the denominator. The two pieces we have had at the beginning is now multiplied by 4. The result we have now, as seen in the illustration above, is 8/15. The process we have described in this paragraph is illustrated below. The 8 pieces are highlighted in yellow. The total amount of pieces of the chocolate bar (15) are shown in purple. The original total amount of three pieces in the chocolate bar is represented by the pink lines (3 lines). The original two pieces in the fraction is highlighted in yellow (2 pieces).

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Blue Ocean Strategy

The blue ocean strategy, written by Kim and Mauborgne, is a market strategy to look for new waters: target new markets - hence the name blue ocean strategy. The opposite of the blue ocean strategy, is the red ocean strategy. Which is bringing your products in a already established market to compete with your competitors. With the blue ocean strategy, however, a company would try to attract a new market by introducing innovating products to beat the current market. 

To give an example in the current game console industry: late director of Nintendo: Iwata, is known for applying the blue ocean strategy in his business. Instead of competing with Microsoft’s Xbox and Sony’s Playstation, by just introducing a new game system with higher graphics, Nintendo always pursued to target new audience by introducing an innovating product. This year, for example, Nintendo released the Nintendo Switch. A highly portable console with multiple controllers that are integrated to a small system. Users are able to play multiplayers game on one system, on the go. The Nintendo Switch was a success, and reached numerous sales records within weeks of release. Nintendo therefore looked for a way to introduce a new, innovative product in the market to beat the products of Microsoft and Sony that are stagnating in terms of innovation.

To summarize this strategy: Stop attempting to beat the competion and start focussing on how to make the competition irrelevant.

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Multiplying Monomials with Polynomials

When multiplying monomials with polynomials you simply have to distribute the monomial to the polynomials.

This is demonstrated in the example taken from khanacademy:

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Multiplying & Simplifying Monomials

By multiplying monomials with each other we can simplify two monomials. This is demonstrated in the following example:

Question 1: Simplify the following two monomials:

(7x^3)(-3x^8)

Answer: 

1. Firstly, split all the coefficients, exponents and common ratios in the example:

(7)(x^3)(-3)(x^8)

2. Secondly, rearrange all the parts:

(7)(-3)(x^3)(x^8)

3. Lastly, put all the parts together. Since multiplication is commutative, they can be easily rearranged into one monomial:

-21x^11

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Multiplying Monomials

Today we're going to take a look at multiplying monomials. From the knowledge you have accumilated thus far, this should be really simple. Let's take a look at the following example:

5x^2*4x^3

The answer for the example is 20x^5 The reason is very simple. Let's explain this by writing the example out:

5 * x * x * 4 * x * x * x

This can be put in order as the following:

5 * 4 * x * x * x * x * x

This can be further simplified by multiplying 5 and 4 with each other. The only thing that need to done is of course adding the remaining x's with each other. The answer is therefore:

20x^5