You notable products they are algebraic expressions used in many mathematical calculations, for example, in first- and second-degree equations.
The term "remarkable" refers to the importance and notability of these concepts for the field of mathematics.
Before we know its properties, it is important to be aware of some important concepts:
- square: raised to two
- cube: raised to three
- difference: subtraction
- product: multiplication
Properties of Notable Products
Square of the sum of two terms
O sum square of the two terms is represented by the following expression:
(a + b)2 = (a + b). (a + b)
Therefore, when applying the distributive property we have to:
(a + b)2 = the2 + 2ab + b2
Thus, the square of the first term is added to the double of the first term by the second term, and finally, added to the square of the second term.
Two-Term Difference Square
O difference square of the two terms is represented by the following expression:
(a - b)2 = (a – b). (a - b)
Therefore, when applying the distributive property we have to:
(a - b)2 = the2 - 2ab + b2
Hence, the square of the first term is subtracted by double the product of the first term by the second term, and finally added to the square of the second term.
The Product of the Sum of the Difference of Two Terms
O product of sum for difference two terms is represented by the following expression:
The2 - B2 = (a + b). (a - b)
Note that when applying the distributive property of multiplication, the result of the expression is the subtraction of the square of the first and second terms.
The Cube of the Sum of Two Terms
O sum cube of two terms is represented by the following expression:
(a + b)3 = (a + b). (a + b). (a + b)
Therefore, when applying the distributive property we have:
The3 + 3rd2b+3ab2 + b3
In this way, the cube of the first term is added to the triple of the product of the square of the first term by the second term and the triple of the product of the first term by the square of the second term. Finally, it is added to the cube of the second term.
The Two-Term Difference Cube
O difference cube of two terms is represented by the following expression:
(a - b)3 = (a – b). (a – b). (a - b)
Therefore, when applying the distributive property we have:
The3 - 3rd2b+3ab2 - B3
Thus, the cube of the first term is subtracted by the triple of the product of the square of the first term by the second term. Therefore, it is added to the triple of the product of the first term and the square of the second term. And finally, it is subtracted to the cube of the second term.
Entrance Exam Exercises
1. (IBMEC-04) The difference between the square of the sum and the square of the difference of two real numbers is equal:
a) the difference of the squares of the two numbers.
b) the sum of the squares of the two numbers.
c) the difference of the two numbers.
d) double the product of the numbers.
e) four times the product of numbers.
Alternative e: to quadruple the product of numbers.
2. (FEI) Simplifying the expression shown below, we obtain:
a) a + b
b) a² + b²
c) ab
d) a² + ab + b²
e) b - a
Alternative d: a² + ab + b²
3. (UFPE) If x and y are distinct real numbers, so:
a) (x² + y²)/(x-y) = x+y
b) (x² - y²)/(x-y) = x+y
c) (x² + y²)/(x-y) = x-y
d) (x² - y²)/(x-y) = x-y
e) None of the above alternatives are true.
Alternative b: (x² - y²)/(x-y) = x+y
4. (PUC-Campinas) Consider the following sentences:
I. (3x - 2y)2 = 9x2 - 4y2
II. 5xy + 15xm + 3zy + 9zm = (5x + 3z). (y + 3m)
III. 81x6 - 49th8 = (9x3 - 7th4). (9x3 + 7th4)
a) I is true.
b) II is true.
c) III is true.
d) I and II are true.
e) II and III are true.
Alternative e: II and III are true.
5. (Fatec) The true sentence for any numbers The and B real is:
a) (a - b)3 = the3 - B3
b) (a + b)2 = the2 + b2
c) (a + b) (a - b) = a2 + b2
d) (a - b) (a2 + ab + b2) = the3 - B3
and the3 - 3rd2b+3ab2 - B3 = (a + b)3
Alternative d: (a - b) (a2 + ab + b2) = the3 - B3
Read too:
- Notable Products - Exercises
- Polynomials
- Factorization
- Algebraic Expressions
- Exercises on Algebraic Expressions
