26 November 2023

during our learning in high school and early college, there are issues that are essential for the student to know. The law of signs, complex numbers, simplification methods and so on are just some of the things that are often seen in linear algebra and mathematics as such. The great detail of this is that not all people have the means or the ability to understand the relevance of the calculation.

So, for this opportunity, we want to focus on what is known as a perfect square trinomial or by its acronym TCP. This is part of elementary algebra, which can be of great use to you in future math and numerical science classes later. Today you will know what a TCP is, as well as other very interesting data that is very useful for you.

What does TCP mean in mathematics?

Any polynomial that has three terms is considered a perfect square trinomial, as long as two of its terms are perfect squares. In addition, for a TCP to exist, such a condition must be met, in addition to the fact that the third term must have to be one that contains the base of double perfect squares. It may be a bit difficult for you to understand this, but let’s look at the following example: 36x^2 + 12xy^2 + y^4. If we break it down by part, we have the following points to consider:

  1. The first term is square of 6x, since, if you multiply it by it, it will give you as a result 36x^2.
  2. The last term also satisfies this condition, since y^2 doing a multiplication by itself, gives us and^4.
  3. The second term in the equation satisfies the special condition mentioned. Its base is the product of double perfect squares. In such a way, that if we multiply 2 (double) by 6xy^2, the result is the term 12xy^2.

It is vital that you always keep in mind the like terms that you have in a polynomial. The reason for this is because you won’t always run into math problems that only have three terms. You will most likely come across some where there will be more than 5 terms. Your goal is to verify if it can be simplified through mathematical procedures and basic algebra. If so, in the end, you will be able to identify if you are facing a perfect square trinomial or not.

Keep in mind that in order for a perfect square trinomial to exist and you can apply the calculus of variables, it has to have positive perfect squares, while the term of the double product can be negative or positive. If you meet all these conditions, you can apply mathematics to solve iteither with factorization or any other method of simplification.

How do you get the TCP in math?

The first alternative is to identify if you are facing a TCP or if you have to take the polynomial to an expression that allows you to do this. Assuming that it is the first case, you will have to do the following:

  • See if the first and third terms have variables and powers.
  • These two mentioned terms have to be perfect squares if you decide to join them.
  • If both terms that make up the perfect square have an exact square root, the condition still holds.
  • Both terms have to be positive.
  • the third term can be a positive or negative numberplus it has to be twice the product of the perfect square binomial.

Now, as far as how to get a TCP, there are two easy ways. The first is through the algebraic development of a mathematical expression like the following: (3 + x)^2. By developing it, we obtain the following: a^2 + 2ab + b^2 = (a + b) ^2. As you can observe, has gone from being a perfect square binomial to being a trinomial. The second way is completely the opposite of what we have shown and explained to you. So you will start from an expression of three terms and you will take it to one of two. This is the basic theory of a perfect square trinomial applied to math exercises.

What is the formula to get a TCP?

If all of the above has seemed a bit complex to understand, you can apply the definition or the TCP formula. As such it is a single formula, but with a slight difference in the operative sign of the second term. Thus, the base formula is as follows: a^2 + 2ab + b^2, however, there are cases where the sign of the middle term is negativeso that expression would look like this: ^2 – 2ab + b^2.

This little formula will help you greatly to identify if you are in the presence of a TCP in a math exercise. So it becomes much easier to learn what a TCP is and how to solve it. It is something as simple as taking each term and applying it to the formula. If it complies with the same structure, then it is a perfect square trinomial. Keep in mind that algebra allows you to play with mathematical expressions, so you can arrange the polynomial to be solved in order to obtain something similar to the base formula.

What is obtained by factoring a TCP?

In order to factor a perfect square trinomial, it is necessary that the first and third terms have perfect square roots. That is to say, that they have an even index and that they are numbers that when multiplied by itself, result in the expression of the polynomial. In addition, the second term or the middle term has to be the result of said multiplication. For the rest, there is nothing else to do, but to follow the theoretical bases that we have given you so far.

TCP exercises in mathematics

A basic exercise that you can use to practice and identify the perfect square trinomials, are the following that we will show you below:

  • x^2 − 2x + 1 =?
  • x^2 + 10x + 25 =?
  • 16x^2 – 40x + 25 =?

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