Productsremarkable are multiplications where the factors are polynomials. There are five most relevant notable products: sum square, difference square, sum product by difference, sum cube and difference cube.
sum square
The products between polynomials known as squares gives sum are the type:
(x + a)(x + a)
The name sum square is given because the representation by potency of this product is as follows:
(x + a)2
The solution for this productremarkable will always be the polynomial Next:
(x + a)2 = x2 + 2x + a2
This polynomial is obtained by applying the distributive property as follows:
(x + a)2 = (x + a)(x + a) = x2 + xa + ax + a2 = x2 + 2x + a2
The end result of this productremarkable can be used as a formula for any hypothesis where there is a sum squared. Generally, this result is taught as follows:
The square of the first term plus twice the first times the second plus the square of the second term
Example:
(x + 7)2 = x2 + 2x7 + 49 = x2 + 14x + 49
Note that this result is obtained by applying the distributive property to (x + 7)
2. Therefore, the formula is obtained from the distributive property over (x + a)(x + a).difference square
O square gives difference The following is:
(x - a) (x - a)
This product can be written as follows using power notation:
(x - a)2
Your result is as follows:
(x - a)2 = x2 – 2x + a2
Realize that the only difference between the results of the square gives sum and of the difference is a minus sign at the middle term.
Generally, this remarkable product is taught in the following way:
The square of the first term minus twice the first times the second plus the square of the second term.
Do not stop now... There's more after the advertising ;)
product of sum for difference
It's the productremarkable which involves a factor with an addition and another with a subtraction. Example:
(x + a)(x - a)
There is no representation in the form of potency for this case, but its solution will always be determined by the following expression, also obtained with the technique of square gives sum:
(x + a)(x - a) = x2 - a2
As an example, let's calculate (xy + 4)(xy – 4).
(xy + 4)(xy - 4) = (xy)2 – 162
That productremarkable is taught as follows:
The square of the first term minus the square of the second term.
sum cube
With the distributive property, it is possible to create a "formula" also for products with the following format:
(x + a)(x + a)(x + a)
In power notation, it is written as follows:
(x + a)3
By means of the distributive property and simplifying the result, we will find the following for this productremarkable:
(x + a)3 = x3 + 3x2at + 3x2 + the3
So, instead of doing an extensive and tiring calculation, we can calculate (x + 5)3, for example, easily as follows:
(x + 5)3 = x3 + 3x25 + 3x52 + 53 = x3 + 15x2 + 75x + 125
difference cube
O cube gives difference is the product between the following polynomials:
(x – a)(x – a)(x – a)
Through the distributive property and simplifying the results, we will find the following result for this product:
(x - a)3 = x3 – 3x2at + 3x2 - a3
Let's calculate the following as an example cube gives difference:
(x - 2y)3
(x - 2y)3 = x3 – 3x22y + 3x (2y)2 – (2y)3 = x3 – 3x22y + 3x4y2 – 8y3 = x3 – 6x2y + 12xy2 – 8y3
By Luiz Paulo Moreira
Graduated in Mathematics
Would you like to reference this text in a school or academic work? Look:
SILVA, Luiz Paulo Moreira. "What are notable products?"; Brazil School. Available in: https://brasilescola.uol.com.br/o-que-e/matematica/o-que-sao-produtos-notaveis.htm. Accessed on June 27, 2021.
