Product of the sum of two monomials by their difference

In this lesson, we will focus on the algebraic identity of the product of the sum of two monomials and their difference. This significant algebraic identity is of significant importance because it allows simplifying the expression of many equations, reducing the number of terms, and facilitating the resolution of complex mathematical problems.

Furthermore, we will see examples of this notable algebraic identity and provide a geometric interpretation of it.

Product of the Sum of Two Monomials and Their Difference

Suppose we have two binomials composed of the same monomials, which we indicate with $A$ and $B$ , where the only difference is the sign of $B$ :

\text{binomial 1: } A + B

\text{binomial 2: } A - B

If we try to multiply the first binomial by the second, we get:

\left( A + B \right) \cdot \left( A - B \right) =

= \quad A \cdot A - A \cdot B + B \cdot A - B \cdot B

However, the terms $A \cdot B$ and $B \cdot A$ are like terms, so they can be subtracted, and the final product is the binomial:

= \quad A^2 - B^2

This result is an important algebraic identity:

Definition

Product of the Sum of Two Monomials and Their Difference

Multiplying the sum of two monomials by their difference results in the product of the squares of the two monomials:

\left( A + B \right) \cdot \left( A - B \right) = A^2 - B^2

Examples

Let's look at some practical examples.

Example

Example 1

\left( 2x + 3 \right) \cdot \left( 2x - 3 \right) =

= \quad 2x \cdot 2x - 2x \cdot 3 + 3 \cdot 2x - 3 \cdot 3

= \quad 4x^2 - 6x + 6x - 9

= \quad 4x^2 - 9

Example

Example 2

\left( 2x^2 + 3x \right) \cdot \left( 2x^2 - 3x \right) =

= \quad 2x^2 \cdot 2x^2 - 2x^2 \cdot 3x + 3x \cdot 2x^2 - 3x \cdot 3x

= \quad 4x^4 - 6x^3 + 6x^3 - 9x^2

= \quad 4x^4 - 9x^2

Example

Example 3

\left( 3a + 2b^3 \right) \cdot \left( 3a - 2b^3 \right) =

= \quad 3a \cdot 3a - 3a \cdot 2b^3 + 2b^3 \cdot 3a - 2b^3 \cdot 2b^3

= \quad 9a^2 - 6ab^3 + 6ab^3 - 4b^6

= \quad 9a^2 - 4b^6

Geometric Interpretation

A very simple geometric interpretation can be given to the product of the sum of two monomials and their difference.

For simplicity, let's focus on the case where the two monomials are $x$ and $y$ and assume that $x > y$ .

Consider a square with side $x$ as shown in the following figure. Its area will be $x^2$ .

**Picture 1:** Area of a square with side x

Now, cut out a square with side $y$ from the upper left corner as shown in the next figure. The area of the cut-out square is $y^2$ . The area of the remaining figure is $x^2 - y^2$ .

**Picture 2:** Cutout of a square with side y

Now, if we cut the figure with area $x^2 - y^2$ as shown in the following figure:

**Picture 3:** Area of the Resulting Figure

and move the top rectangle by rotating it to the right, we get a rectangle with area $x^2 - y^2$ :

**Picture 4:** Geometric interpretation of the product of the sum of two monomials and their difference

However, the sides of the rectangle are equivalent to $x + y$ and $x - y$ . Therefore, the product of the sum of two monomials and their difference is equal to the area of the shown rectangle.

In Summary

In this lesson, we have examined another important algebraic identity that frequently appears in solving mathematical problems: the product of the sum of two monomials and their difference.

\left( A + B \right) \cdot \left( A - B \right) = A^2 - B^2

We have seen how this algebraic identity can be geometrically interpreted as the area of a rectangle resulting from cutting out a square whose side equals the monomial with a negative sign from the square whose side equals the monomial with a positive sign.