Square of a Trinomial
The square of a trinomial is a product between polynomials that falls into the category of significant Algebraic Identities. It allows for the quick calculation of the multiplication of a trinomial by itself.
Through its application, it is possible to reduce the square of a trinomial to a polynomial composed of six terms: three squares and three double products.
In this lesson, we will derive the general formula for the square of a trinomial and try to apply it to some examples.
Square of a Trinomial
Squaring a trinomial means multiplying it by itself. Thus, we can replace the square with the product of the trinomial by itself as follows:
In this case, in place of
Let's try to derive the final formula by expanding the polynomial product:
In the result above, we have various similar monomials that can be summed up:
So, the final result is:
Square of a Trinomial
The square of a Trinomial:
is a polynomial composed of 6 terms:
- The 3 squares of individual terms:
, , and . - The 3 double products of each term with another:
, , and .
Examples
Let's now apply the formula for squaring a trinomial to some examples.
Example 1
Let's calculate the square of the following trinomial:
Using the formula, label the 3 terms of the trinomial with
Replace the three terms in the formula:
Now, let's confirm the result by directly expanding the product of the polynomials:
At this point, sum up the similar monomials:
And the result is confirmed.
Example 2
Let's now expand the square of a more complex trinomial which also has negative terms:
As before, label the three terms that make up the trinomial with
Now apply the trinomial square formula by replacing the three terms in
Example 3
Expand the following trinomial square:
Also in this case, let's set:
Then, apply the formula, paying attention to the signs:
$$ = (-3a^3)^2 + \left( \frac{3}{2}a \right)^2 + (-2)^2 + 2(-3a^3) \left( \frac{3}{2}a \right) + 2(-3a^3)(-2) + 2\left( \frac{3}{2}a \right
)(-2) $$
In Conclusion
In this lesson, we derived the formula for the direct expansion of the square of a trinomial. The result is a polynomial of six terms made up of three squares and three double products.
Specifically, the square of a trinomial consists of the three squares of the individual terms and the three double products of each term with the other.
However, when applying this notable product, one must pay attention to the signs of the individual terms. Indeed, during the substitution phase in the formula, it's important to remember to carry over the sign to avoid making mistakes.