Explaining Math: Exponential Functions, Strategies for Factoring and Square of a Trinomial

Many people are good at math, but not everyone likes it. In this post, let's make it easier to understand. To do it, three of the countless General Mathematics topics - the exponential functions, getting the square of a trinomial, and the Factoring of Trinomials, are explained concisely and understandably.

For discussion, several illustrations are used to explain the process further. Exponential functions come first. This article has explained its definition and its difference from other terms, provides examples, and then does the same for strategies on factoring and getting the square of perfect trinomials.

Exponential Functions

It is a function in which the value is constantly raised to the power of the argument, especially the function where the constant is e.

a is any constant number

An exponential function with base b is a function of the form f(x) = b^x or y = b^x (where b > 0, b ≠ 1).

Natural Exponential Function          

It is a function defined by e- an accurate to eight decimal places 2.71828183 and called “Euler number” in honor of the mathematician Leonhard Euler.

Exponential Equation

If two powers with same base are equal, then their exponents must be equal. ( Property of equality).

Example:

Exponential Inequality

          It is an Inequality involving exponential expressions.  (Property of Inequality).

Example:

Here's another topic, and it is about strategies in factoring.

Student learners have varied needs, interests and abilities. They go to school with different states and experiences in life. Developing one's numerical proficiency may start by understanding real-life situations such as problem-solving situations.

Strategies for  Factoring

Difference of Squares

a² - b²  =  (a- b) (a+ b)

Square of Sum

a² + 2ab + b²  =  (a + b)²  

Square of Difference

a² - 2ab + b²  =  (a - b)²

^{Factoring examples:}

25c² - 9d²

*In this example, there were only two terms and it is an example of a Difference of Squares. 

So get the square root of the 1st and 2nd term

5c and 3d

Next is make it as a Difference of Squares

=(5c-3d)(5c+3d)

6e³ - 24e² + 24e

*Get the Greatest Common Factor first and bring it out.

6e (e² - 4e + 4) 

Since (e² - 4e + 4) is a perfect square you can now get the factors by using the Square of Difference.

=6e(e - 2)²   

   

Why not try another problem on your own? Just do it and comment your answer. No one will judge you if you get it wrong. Instead, we will help you understand it further.

Try the following exercises:

x² + 6x + 9

c² + 8cd + 16d²

-16g² +6g²

z² -81

8-6y+y²

Quick Steps in Getting the Square of a Trinomial

1. You must get first the sum of the squares of the 1^st, 2^nd and 3^rd terms;

2. Then, double the product of the 1^st and the 2^nd terms;

3. Next in line is to double the product of the 1^st and the 3^rd terms; and

4. Lastly, you need to double the product of the 2^nd and 3^rd terms.

Study the example below:

(a – b + c)²

Step 1. Sum of the squares of the 1^st, 2^nd and 3^rd terms.

Since the 1^st, 2^nd and 3^rd terms are a, -b, and c then the sum of their squares  would be:

a² + b² + c² 

Step 2. Double the product of the 1^st and the 2^nd terms.

The 1^st and  2^nd are a and -b. After doubling their product.

This is what you will get:

-2ab

Step 3. Double the product of the 1^st and the 3^rd terms.

The 1^st and  3^rd  are a and c. After doubling their product.

This is what you will get:

+2ac

Step 4. Double the product of the 2^nd and the 3^rd terms.

The 2^nd and  3^rd  are -b and c. After doubling their product.

This is what you will get:

-2bc

Now, just form the answers from Step 1 to Step 4 to get the final answer.

Step 1. a² + b² + c² 

Step 2. -2ab

Step 3. +2ac

Step 4. -2bc

= a² + b² + c²-2ab+2ac-2bc

Try answering the following:

(x+y+z)²

(m + a – y) (m + a – y)

(4a + 4b + 4c)²

(1.5j2 – 2.3u + 1)²

(2g² + 3h – 5i)²

 (j + 2u + l)²

(3a + 2b – 1c)²

(j + o – 2k)

That is all for now. Again, General Mathematics has a lot of topics, and three of the most fun topics are just discussed in this post. Hopefully, you gained something valuable. Remember that knowledge is power, so invest in more learning. 

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