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
If we try to multiply the first binomial by the second, we get:
However, the terms
This result is an important algebraic identity:
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:
Examples
Let's look at some practical examples.
Example 1
Example 2
Example 3
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
Consider a square with side
Now, cut out a square with side
Now, if we cut the figure with area
and move the top rectangle by rotating it to the right, we get a rectangle with area
However, the sides of the rectangle are equivalent to
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.
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.