Square of a Binomial

The square of a binomial is a algebraic identity that allows for a quick transformation of a binomial multiplied by itself, thus squared, into a trinomial.

When applying the rule of the square of a binomial, attention must be paid to the signs with which the terms appear in the binomial itself. Specifically, care must be taken as to whether the two terms of the binomial have the same or opposite signs.

In this lesson, in addition to showing how to apply it, we will also see the proof of the formula for the square of a binomial and give a geometric interpretation of it. Lastly, we will also look at some practical examples.

Square of a Binomial

To square a binomial means to multiply the binomial by itself. Thus, the square of a binomial can be replaced by a simple product:

Whatever the monomials and may be.

Expanding the product to the right of the equals sign, we get:

But the terms and are like terms, so they can be combined, and the final product is the trinomial:

Let's see an example:

Square of a Binomial with a Difference

When applying the formula for the square of a binomial, one must pay attention to the signs of the involved monomials.

Let's try to compute the square of the following binomial:

Compared to the previous example, the second monomial of the binomial has a negative sign. There's no difference in applying the formula for the square of a binomial, but one needs to be cautious about the signs. In fact, when applying the formula, we get:

This expression becomes:

One should note, in the resulting trinomial, the negative sign of the double product of and : . In fact, the terms and in the trinomial are squared, and hence they will always appear with a positive sign in the outcome. Conversely, the term will display a sign that depends on whether and have opposite signs in the starting binomial. In such a case, we can rewrite the formula in this way:

If had had a negative sign instead of , there would be no difference. In fact, in this case, it suffices to change the order of the two terms:

and the end result would always be:

which is identical to the result above.

Lastly, we wonder: "what if both and were negative?"

Let's derive the result:

Applying polynomial multiplication, we get:

In this case, the sign of the double product of the two monomials is positive. Thus:

Geometric Interpretation

For the square of a binomial, we can provide a geometric interpretation. In fact, if we take the square with the side of the binomial , it follows that its area is equal to the square of the binomial in question.

Let's take a close look at the following figure:

We can see that the area of the square with the side consists of the sum of the areas of two squares and plus the sum of the areas of the rectangles in red and yellow. But the area of the red and yellow rectangles is the same and is equal to , so the sum of their areas is .

Essentially, the total area of the square can be expressed by the sum:

which is the formula for the square of a binomial that we derived earlier.

Examples

Let's look at a few more examples.

In Conclusion

In this lesson, we derived the first significant Algebraic Identity regarding the product between polynomials: the square of a binomial. When we are dealing with a binomial multiplied by itself, we can directly apply the formula found without having to resort directly to the product between polynomials.

However, when applying the formula, care must be taken with the signs of the two terms of the binomial. Indeed, if these signs are concordant, meaning both positive or both negative, the result of the square of the binomial has the intermediate term with a positive sign:

Conversely, if the two terms have discordant signs, the square of a binomial has the intermediate term with a negative sign: