Square Trinomial Example

On algebra, a trinomial is an expression that has three terms, that is, three values ​​that are being added or subtracted. They result from operations such as the square of a binomial, in which, when the terms are added to each other (adding or subtracting them), three remain different variables. An example of a trinomial is the following:

x² + 2xy + y²

In this trinomial, three terms are noted: (x²), (2xy), (Y²), and between them are plus signs (+). They are written like this because can no longer be reduced. This means that they cannot be added between them so that two or one term remains.

How do you get a trinomial?

The simplest way a trinomial can be obtained is with one of the remarkable products: the binomial squared. The operation happens as follows:

If the binomial is:

x + y

The rule to solve it is:

• Square of the first term (x * x = x²)
• Plus the double product of the first times the second + (2 * x * y = 2xy)
• Plus the square of the second + (y * y = Y²)

The result is the following trinomial:

x² + 2xy + y²

This is called Perfect square trinomial. Pay attention: there are two concepts that must be learned to differentiate correctly:

• Perfect square trinomial: It is the result of a squared binomial.
• Trinomial squared: It is a trinomial that multiplies by itself, that is, it is squared.

Trinomial squared example

The trinomial squared is an algebraic operation in which a trinomial multiplies by itself to be squared. The procedure to obtain it is to multiply term by term, until obtaining those that are going to form the result.

For the same trinomial from the beginning:

x² + 2xy + y²

The operation is written:

(x² + 2xy + y²) ²

Which is the same as:

(x² + 2xy + y²) * (x² + 2xy + y²)

Procedure to calculate it

A very simple way to develop the operation will be established, which consists of multiply all the trinomial for each of the terms. It is explained:

Step 1: (the whole trinomial) * (first term)

(x² + 2xy + y²) * x²

One by one:

(x²) * x² = x⁴

(2xy) * x² = 2x³Y

(Y²) * x² = x²Y²

Results of Step 1:

x⁴ + 2x³y + x²Y²

Step 2: (the whole trinomial) * (second term)

(x² + 2xy + y²) * 2xy

One by one:

(x²) * 2xy = 2x³Y

(2xy) * 2xy = 4x²Y²

(Y²) * 2xy = 2xy³

Results of Step 2:

2x³and + 4x²Y² + 2xy³

Step 3: (the whole trinomial) * (third term)

(x² + 2xy + y²) * Y²

One by one:

(x²) * Y² = x²Y²

(2xy) * and² = 2xy³

(Y²) * Y² = and⁴

Results of Step 3:

x²Y² + 2xy³ + and⁴

Step 4: The three results are added

Results Step1: x⁴ + 2x³y + x²Y²

Results Step 2: 2x³and + 4x²Y² + 2xy³

Results Step 3: x²Y² + 2xy³ + and⁴

Sum: x⁴ + 2x³y + x²Y² + 2x³and + 4x²Y² + 2xy³ + x²Y² + 2xy³ + and⁴

Step 5: Similar terms are reduced

x⁴ + 2x³y + x²Y² + 2x³and + 4x²Y² + 2xy³ + x²Y² + 2xy³ + and⁴

x⁴ + 2 (2x³y) + 6 (x²Y²) + 2 (2xy³) + and⁴

x⁴ + 4x³and + 6x²Y² + 4xy³ + and⁴

Law for the squared trinomial

If it is required to establish a law to calculate the trinomial squared based on the result obtained, it would be written like this:

Square of the first term

Plus the double product of the first times the second

Plus six times the product of the first by the third

Plus the double product of the second times the third

Plus the square of the third

Be part of the example. The trinomial is:

x² + 2xy + y²

The result has been:

x⁴ + 4x³and + 6x²Y² + 4xy³ + and⁴

• Follow with: Trinomial cubed.