Abbreviated Multiplication Formulas

Abbreviated multiplication formulas are here to stay, and we will use them in almost all the exercises we encounter in the future.
But hey! Don't stress out. We are lucky to know them because they are the ones that will help us solve exercises correctly and efficiently.
We will separate the three abbreviated multiplication formulas into $4$ categories:

Multiplication of the Sum and Difference of Two Terms
$(X + Y)\times (X - Y) = X^2 - Y^2$
Difference of Squares Formula
$(X - Y)^2=X^2 - 2XY + Y^2$
Sum of Squares Formula
$(X + Y)^2=X^2+ 2XY + Y^2$
Formulas related to two expressions in the $3rd$ power
$(a+b)^3=a^3+3a^2 b+3ab^2+b^3$
$​​​​​​​(a-b)^3=a^3-3a^2 b+3ab^2-b^3$

Let's start with the first category:

$(X + Y)\times (X - Y) = X^2 - Y^2$

As you can see, this formula can be used when there is a product between the sum of two specific terms and the difference of those two terms.
Instead of presenting them as a product between a sum and a difference, you can write it $X^2 - Y^2$.
Similarly, if you are presented with such an expression $X^2 - Y^2$ that represents the difference of two squared numbers, you can write it like this: $(X + Y)\times (X - Y)$
Pay attention: it says in the formula $X$ and $Y$ But it works both in non-algebraic expressions and in expressions that combine variables and numbers.

If we are given:
$(x+2)(x-2)$
We can see that it is a product of the sum of the two terms and the difference of the two terms.
Therefore, we can express the same expression according to the formula in the following way:
$x^2-2^2$
$x^2-4$
Similarly, if we were given the expression:
$x^2-4$
we could express $4$ as a square number, that is $2^2$,
to arrive at a form that matches the formula:
$x^2-2^2$
and, therefore, use the formula and express the expression in this way:

$x^2-2^2=(x-2)(x+2)$

Excellent.
Now let's move on to the formula for the difference of squares.

$(X - Y)^2=X^2 - 2XY + Y^2$

This formula describes the square of the difference between two numbers, that is, when we encounter two numbers with a minus sign between them, the difference, and they will be in parentheses and appear as an expression squared, we can use this formula.
Keep in mind that, although the formula contains algebraic elements, the formula also works with non-algebraic expressions or combined expressions between numbers and algebra.

$(X-5)^2=$
Here we identify two terms with a minus sign between them, enclosed in parentheses and raised to the square as a single expression.
Therefore, we can use the difference of squares formula.
We will work according to the formula and pay attention to the minus and plus signs.
We will obtain:
$(X-5)^2=x^2-10x+25$
Basically, we have expressed the same expression differently using the formula.
Great. Now let's move on to the sum of squares formula.

$(X + Y)^2=X^2+ 2XY + Y^2$

This formula describes the square of the sum of two numbers, that is, when we encounter two numbers with a plus sign in the middle, meaning a sum, and they are enclosed in parentheses and appear as a squared expression, we can use this formula.
Keep in mind: although algebraic elements appear in the formula, it also works with non-algebraic expressions or combined expressions between numbers and algebra.
Note: this formula is very similar to the formula for the difference of squares and differs only in the minus sign in the middle term.

$(X+4)^2=$
Here we identify two terms with a plus sign between them, enclosed in parentheses and raised to the square as a single expression.
Therefore, we can use the formula for the sum of squares.
We work according to the formula and pay attention to the minus and plus signs.
We will obtain: 
$(X+4)^2=x^2+8x+16$
Basically, we express the same expression differently using the formula.

Now, after you have become deeply familiar with the previous abbreviated multiplication formulas of $2°$ degree, we will move on to the formulas related to two expressions to the 3rd power.

The formulas that refer to two expressions raised to the power of $3$

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

Here we can also recognize that there are two different formulas for the difference and the sum of terms.
Let's start with the first formula for the sum:
$(a+b)^3=a^3+3a^2 b+3ab^2+b^3$
Here we can also recognize that there are two different formulas for the difference and the sum of terms.
Let's start with the first formula for the sum:

When we are given the following expression:
$(X+2)^3=$
We can identify two terms with a plus sign between them that are enclosed 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.
$(X+2)^3=x^3+3\times x^2\times 2+3\times x\times 2^2+2^3$
$(X+2)^3=x^3+6x^2+12x+8$
Basically, we express the same expression differently with the help of the formula.

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

$​​​​​​​(a-b)^3=a^3-3a^2 b+3ab^2-b^3$

This formula describes a way to express the difference of two terms, when they are enclosed in parentheses and raised as a single expression to the power of three.
The formula can be used with algebraic terms or with numbers and also in combination.

When we are given the following expression:
$(X-4)^3=$
We can identify two terms with a minus sign between them that are enclosed 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. Pay attention to the minus and plus signs.
$(X-4)^3=x^3-3\times x^2\times 4+3\times x\times 4^2-4^3$
$(X-4)^3=x^3-12x^2+48x-64$
Basically, we express the same expression differently using the formula.

$(X + 2)^2=X^2 - 8$

$-(X + 2)^2=-X^2 - 8$

$(X + 3)^2=(X-4)\times (X+4)$

$(X + 2)^2=X^2 - 8$

$-(X + 2)^2=-X^2 - 8$

$(X + 3)^2=(X-4)\times (X+4)$

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

Positive, Negative Numbers and Zero

The Real Number Line

Opposite Numbers

Absolute Value

Elimination of Parentheses in Real Numbers

Multiplication and Division of Real Numbers

Multiplicative Inverse

Multiplication Tables

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

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