Abbreviated Multiplication Formulas Practice Problems

What are Abbreviated Multiplication Formulas?

Abbreviated multiplication formulas, also known as algebraic identities, are shortcuts that simplify the process of expanding and factoring expressions. These formulas save time and reduce the steps required for complex calculations. Abbreviated multiplication formulas will be used throughout our math studies, from elementary school to high school. In many cases, we will need to know how to expand or add these equations to arrive at the solution to various math exercises.

Here are the basic formulas for abbreviated multiplication:

The square of the sum:
$(X + Y)^2=X^2+ 2XY + Y^2$

The squared difference:

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

The difference of squares:

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

Abbreviated multiplication formulas for 3rd-degree expressions, also known as cubic identities, build upon the concepts of the 2nd-degree formulas we’ve already covered. The key difference is the adjustment for working with cubic (3rd-degree) terms instead of quadratic (2nd-degree) terms. These formulas simplify complex cubic expressions, breaking them into manageable parts to make calculations faster and more efficient. They are particularly useful in solving problems involving volumes of cubes and other 3D shapes or in advanced mathematics, such as polynomial factoring and equation solving.

Here are two of the most common abbreviated multiplication formulas for 3rd-degree expressions:

Cube of a Sum

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

Cube of a Difference

$​​​​​​​(a-b)^3=a^3-3a^2 b+3ab^2-b^3$

We will test the shortcut multiplication formulas by expanding the parentheses.

Square of a Sum:

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

$X^2 + XY + YX + Y^2=$

Since: $XY = YX$

$X^2 + 2XY + Y^2$

Example:

$(x+3)^2$

$(x+3)^2=x^2+2(x)(3)+3^2=x^2+6x+9$

Square of a Difference:

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

$X^2 - XY - YX + Y^2=$

Since:$XY = YX$

$X^2 - 2XY + Y^2$

Example:

$(x−4)^2$

$(x−4)^2=x^2−2(x)(4)+4^2=x^2−8x+16$

Product of a Sum and a Difference:

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

$X^2 - XY + YX - Y^2=$

Since: $XY = YX$

$- XY + YX = 0$

$X^2 - Y^2$

Example:

$(x+5)(x−5)$

$(x+5)(x−5)=x^2−5^2=x^2−25$

Using Abbreviated Multiplication Formulas to Shift the Expression Both Ways

It’s important to remember that these formulas are not one-sided; you can use them to switch between different forms of an expression as needed. For example, you can use the formula:

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

to expand an expression in parentheses into its expanded form. Conversely, if you encounter an expression like

$X^2+ 2XY + Y^2$

you can factor it back into

$(X + Y)^2$

This flexibility allows you to work with the representation that is most useful for the problem at hand, whether it’s simplifying, solving, or analyzing the expression. Understanding this two-way functionality is essential for mastering algebraic manipulation.