Special products: Step by step tutorial. Resolved examples.

If you never remember to apply the special product formulas and when you remember, then you don’t know how to apply it, keep reading because in less than 5 minutes your doubts will be solved.

Multiplying two polynomials together, if they have more than three terms and in their terms have several variables, can be very tedious and you can invest a lot of time in multiplying.

Fortunately, if the polynomials to be multiplied are of two terms (binomials) and their terms are equal, you can apply the formulas of notable products, which will allow you to carry out the multiplication directly.

Would you like to know how to identify when you have a special product and how to apply its formula?

In this post I will explain to you step by step the different special product formulas, with examples resolved step by step, so that you learn how to apply them in your exercises.

Definition os special products

What are special products?

Special products,  are polynomials of two terms (binomials) elevated to the square, or the product of two binomials, as we will see later, whose development always follows the same rules.

Therefore, at the moment you identify them, you directly have to apply their formulas to develop them, without having to multiply term by term, to save you time and complications.

The special product formulas more important are: the square of sum, the square of difference and the sum by difference or difference of squares.

Let’s see every one of them.

Special product formulas

Square of sum formula

The way to read this special product is: Square of the first, plus double of the first by the second, plus the square of the second:

The first i s “a” (the first term of the polynomial) and the second is “b” (the second term of the polynomial).

To apply this formula you only have to replace the terms of the polynomial with a and b. For example:

The first is x and the second is 1. So then:

  • Square of the first: x
  • Double of the first by the second: 2. x. 1
  • Square of the second: 1

So far you’ve applied the formula. Now we have to operate within each term to simplify, multiplying numbers and solving powers:

A serious mistake is not realizing that the above expression is a special product and squaring each term incorrectly:

Square of difference formula

This formula is very similar to the square of sum formula, with the difference of the minus sign in the second term.

It reads as follows: Square of the first, minus twice the first by the second, plus the square of the second:

As in the previous case, the first is “a” (the first term of the polynomial) and the second is “b” (the second term of the polynomial). To apply this formula you only have to replace the terms of the polynomial with a and b.

For example:

  • Square of the first: 2²
  • Double of the first by the second: 2.2.x
  • Square of the second: x²

Both the square of a sum and the square of a difference can be used to factor polynomials, applied in reverse, that is, from the development, obtaining the special products.

Difference of squares formula

This formula is very useful for factoring polynomials and simplifying algebraic fractions when we have the subtraction of two squared terms:

Usually the difference of squares is “camouflaged” and is not directly visible, so it is very important to know how to identify it.

For example: x²-25 is apparently not a difference of squares, because we don’t see the second term elevated to the square, but since we know that 25 is 5² then we can apply the formula:

To be taken into account with the special product formulas

I’ve just shown you the three formulas of the most important special products.

There are other special productssuch as the cube of a sum, the cube of a difference, trinomial squared… that are not at all practical or most complicated to memorize.

In such cases, it is preferable to develop the multiplication of polynomials rather than memorize a formula.

Another thing you have to keep in mind is that the first and second terms don’t have to be made up of a single factor. Each term can consist of several factors, such as:

In that case, be very careful and do not forget to elevate the whole term to the square, with the application of the property of the corresponding power.

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