Formulas for Cubic Expressions

$(a+b)^3=a^3+3a^2 b+3ab^2+b^3$
$(a-b)^3=a^3-3a^2 b+3ab^2-b^3$

These are part of the shortened multiplication formulas and refer only to expressions raised to the power of $3$ .
We can identify that there are two different formulas: one for the subtraction of the elements and another for the addition of the elements.

Let's start with the first formula for addition:

$(a+b)^3=a^3+3a^2 b+3ab^2+b^3$
This formula describes a way to express the sum of two elements, when they are within parentheses and are raised as an expression to the power of three.
Pay attention: the formula is also suitable for use with algebraic elements, numbers, or a combination of them.

Now, let's move on to the second formula for subtraction.

$(a-b)^3=a^3-3a^2 b+3ab^2-b^3$

This formula describes a way to express the sum of two elements, when they are within parentheses and raised as an expression to the power of three.
Pay attention: the formula is also suitable for use with algebraic elements, numbers, or a combination of both.

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Let's look at an example of the sum formula

When we are given the following expression:
$(X+6)^3=$
We can identify two elements with the plus sign, which are in parentheses and raised to the power of three as a single expression.
Therefore, we can use the corresponding formula.
We will work according to the formula and pay attention to the minus and plus signs.
$(X+6)^3=x^3+3\times x^2\times 6+3\times x\times 6^2+6^3$
$(X+6)^3=x^3+18x^2+108x+216$
In reality, we pronounce the same expression differently using the formula.

Let's look at an example of the subtraction formula

When we are given the following expression:
$(X-2)^3=$
We can identify two elements with the minus sign, which are within parentheses and raised to the power of three as a single expression.
Therefore, we can use the corresponding formula.
We will work according to the formula, and pay attention to the minus and plus signs.
$(X-2)^3=x^3-3\times x^2\times 2+3\times x\times 2^2-2^3$
$(X-2)^3=x^3-6x^2+12x-8$
Indeed, we pronounce the same expression differently using the formula.

If you are interested in this article, you might also be interested in the following articles:

Multiplication of the sum of two elements by the difference between them

The formula for the difference of squares

The formula for the sum of squares

In the blog of Tutorela you will find a variety of articles about mathematics.