Trinomial Cubed Example

The trinomial is the algebraic expression that has three terms, with different variables and separated by positive or negative signs. For example: x + 4y - 2z. Among the operations in which it participates, is the trinomial cubed, which is when it is multiplied by itself, obtaining its square, and then the square is multiplied by the same trinomial.

If we take the trinomial as an example x + 4y - 2z, the operation of the trinomial cubed is written like this:

(x + 4y - 2z)³

or like this

(x + 4y - 2z) * (x + 4y - 2z) * (x + 4y - 2z)

The way to solve it is:

• Obtain the square of the trinomial, multiplying term by term
• Multiply the result by the trinomial, again: term to term

• It may interest you: Trinomial squared.

Trinomial cubed example

It is explained, step by step, how to obtain a cubed trinomial:

(x + 4y - 2z)³

 (x + 4y - 2z) * (x + 4y - 2z) * (x + 4y - 2z)

The square of the trinomial is obtained

For him square of a trinomial, multiplies by itself:

(x + 4y - 2z) * (x + 4y - 2z)

The operation is performed by multiplying the terms of the first trinomial for each of the second:

• (x + 4y - 2z) * (x) = x² + 4xy - 2xz
• (x + 4y - 2z) * (4y) = 4xy + 16y² - 8yz
• (x + 4y - 2z) * (- 2z) = -2xz - 8yz + 4z²

Now the results obtained are put together:

x² + 4xy - 2xz + 4xy + 16y² - 8yz - 2xz - 8yz + 4z²

And the similar ones are reduced, leaving six different terms:

x² + 8xy - 4xz - 16yz + 16y² + 4z²

We multiply the square by the trinomial

(x² + 8xy - 4xz - 16yz + 16y² + 4z²) * (x + 4y - 2z)

In this operation, the square is multiplied by the original trinomial, term by term:

• (x² + 8xy - 4xz - 16yz + 16y² + 4z²) * (x) = x³ + 8x²y - 4x²z - 16xyz + 16xy² + 4xz²
• (x² + 8xy - 4xz - 16yz + 16y² + 4z²) * (4y) = 4x²and + 32xy² - 16xyz - 64y²z + 64y³ + 16yz²
• (x² + 8xy - 4xz - 16yz + 16y² + 4z²) * (- 2z) = -2x²z - 16xyz + 8xz² + 32yz² - 32y²z - 8z³

Now the results obtained are put together:

x³ + 8x²y - 4x²z - 16xyz + 16xy² + 4xz² + 4x²and + 32xy² - 16xyz - 64y²z + 64y³ + 16yz² - 2x²z - 16xyz + 8xz² + 32yz² - 32y²z - 8z³

Like terms meet:

x³ + (8 + 4) x²y + (-4 -2) x²z + (-16 -16 -16) xyz + (16 +32) xy² + (4 +8) xz² + (-64 -32) and²z + 64y³ + (16 + 32) and z² - 8z³

x³ + 12x²y - 6x²z - 48xyz + 48xy² + 12xz² - 96y²z + 64y³ + 48yz² - 8z³

The result of the cubed trinomial is:

x³ + 12x²y - 6x²z - 48xyz + 48xy² + 12xz² - 96y²z + 64y³ + 48yz² - 8z³

This has ten terms with different variables, which can no longer be accumulated with each other.