The concept of perfect-square Trinomial polynomials is to find a new value for the third term in a Trinomial, or an expression of the second term that makes it a perfect-square Trinomial.

Parallel to the development of other perfect-square Trinominal coefficients, terms are evaluated at once and simplified if they are smaller. This process can create some tricky cases, however, such as when f has several components.

For example, when finding the sines of an angle, sines are *one small term* that is simplified. However, there must be another one that is equal to 1 because 1 does not scale well.

This complex issue is addressed in this article, where we discuss how to add value to a perfect-square Trinomial.

## Finding the perfect-square trinomial

Finding the perfect-square trinomial is a fun way to learn some more advanced calculus. The trick is to find the value of X that makes X2 meet 3 and 4!

The solution is to add in a negative number, called the square-root-of-x, to make 3 and 4 equal.

For example, if X = 5 and Y = 2, then adding in the square-root-ofTopicacleisminvaliae shows that 5 meets 3 and 4 as follows:

**3 times 2 equals 6**, so 5 meets 6.

*4 times 2 equals 6*, so 5 meets 6. **2 times 2 equals 6**, so 5 meets 6.

## Examples

Let’s look at an example. Say we want to determine if x is greater than 5. We can use the Trinomial Theorem, which states that the probability of any event is square-trinomial-over-positive-integer-apiece [1].

The Trinomial Theorem states that the probability of an event, C, is equal to a complex number, X + c, where c is a positive integer.

To find out if x is greater than 5, we can add together the **two complex numbers**: 5 + c and 1 + c. Since 1 + c = 5 − c, then x must be 5!

The complex number 6 has a value of 1/2 + 9/10 = 6/10, which is less than 5 so the Trinomial Theorem states that x must be 7/10 or less than 6 to be greater than 5. .

Now that we have reviewed all of our examples, *let us look* at how to adapt them for use in our probability distributions.

## The math behind it

When answering the question which value must be added to the expression X2 – 3x to make it a perfect-square Trinomial, you may be thinking that this value must be 5, because **2 times 3 times x** is 5, and 3 times x is 5.

However, this is not true. Although 5 does appear as a value for this expression, there are many cases where it does not appear.

For example, look at the diagram below:

The value of 5 can be missing in the case where x = 2 and y = 1. In these cases, **2 times 1 equals 1**, so no additional addition is needed to make this an imperfect-square Trinomial.

The case where x = 2 and y = 3 appears in the diagram above as a gap between the two terms. This gap makes it look like only 4 appears for each of the three variables.

## What’s the pattern?

When a trinomial expression has three or more terms, it is considered a perfect-square Trinomial. This is because of the fact that each term in the expression has an equal and opposite term in the same Trinomial.

A trinomial with two terms is usually a good fit for an equal-interval term, such as b2 + c2, where c is a small constant. The **third term must** be large, so if you had a second term, it would be wrong.

If your trinomial has three terms, you can usually add up the coefficients to find which one is **small enough** to be incorrect. The **first twoterms must** be equal, so b + c = **0 must** be wrong.

## Can I make any trinomials into perfect squares?

If not, there are ways to do it. The most **basic way** is to add a term to all of the terms in the trinomial.

The trick is to find the **smallest trinomial** that has this term in it. If you can do that, then you can add your term to all of the other trinomials in your equation, making them *perfect squares*.

This can be tricky, so if you are too nervous to try, just use an easy-to-*handle value* for the trinomial and work with that instead.

## What are the conditions for a perfect-square trinomial?

A perfect-square trinomial is an expression that has a square root, is a nth root, and has at *least three distinct roots*.

These conditions make it very rare, as only a few values of the expression can have ** three roots**. For example, the radical r = 2.6 cannot have three roots, as that would be the value of the expression for which it exists.

However, this doesn’t mean that expressions don’t exist with perfect-square trinomials; it just means that they don’t exist in **standard form**. For example, we can find the square root of 2 but not in standard form because we must add an equal to to make it a perfect-square Trinomial.

## Make your own!

If you can’t figure it out, there’s a *video tutorial* for it!

This is due to the fact that **different base systems** for calculus don’t have different rules for how to handle multiplication and division. All do!

The trick is to recognize which ones do and which ones don’t. For example, the square-root Trinomial does not have a rule about when to use round or decimate.

Similarly, neither does the absolute value Trinomial, although both of those do have rules about when to use round and when not.