The Notable Product

The remarkable product is a particular case of the distributive property that gives us the perfect square trinomial as a result and establishes that, if $a$ and $b$ are two real numbers, the square of the sum of them is equal to the first squared plus twice the product of the first times the second plus the second squared, that is,

This equality can be deduced by performing the distributive property when we multiply the sum of two numbers by that same sum, let’s see then,

Similarly, if $a$ and $b$ are two real numbers, the square of the subtraction between the two is equal to the first squared minus twice the product of the first times the second plus the second squared, that is,

This equality can be deduced by performing the distributive property when we multiply the substraction of two numbers by that same substraction, let’s see then,

This type of expression is often found in the development of algebraic operations because we cannot always carry out the sum that is inside the parentheses, let’s see in the following examples how to apply this operation:

Ejemplos

Ejemplo 1

Apply the notable product to expand the expression $(3 + 2)^2$ . We add the two elements within the parentheses and square as follows:

$(3 + 2)^2$
$= 5^2$
$= 25$

Ejemplo 2

Apply the notable product to expand the expression $(3 + \sqrt{2})^2$ . Note that one of the addends involved is the square root of two, therefore it cannot be added with three.

$(3 + sqrt{2})^2$
$= 3^2 + 2(3)(sqrt{2}) + (sqrt{2})^2$
$= 9 + 6sqrt{2} + 2$
$= 11+6sqrt{2}$

Ejemplo 3

Apply the notable product to expand the expression $(\sqrt[3]{6} - 4)^2$ . Note that one of the addends involved is the cube root of six, therefore it cannot be subtracted with four.

$(sqrt[3]{6} - 4)^2$
$= (sqrt[3]{6})^2 -2(sqrt[3]{6})(4) + 4^2$
$= (sqrt[3]{6})^2 -8sqrt[3]{6} +16$

Ejemplo 4

Apply the notable product to expand the expression $(x + 7)^2$ . Note that one of the addends involved is an unknown, therefore it cannot be added with seven.

$(x+7)^2$
$= x^2 + 2(x)(7) + 7^2$
$= x^2 +14x + 49$

Ejemplo 5

Apply the notable product to expand the expression $(2x-8)^2$ . Note that one of the addends involved is an unknown multiplied by two, therefore it cannot be subtracted with eight.

$(2x-8)^2$
$= (2x)^2 - 2(2x)(8) + 8^2$
$= 4x^2 - 32x + 64$

Ejemplo 6

Apply the notable product to expand the expression $(x^2 + x^5)^2$ . Note that one of the addends involved is x squared and the other is x raised to five, therefore they cannot be added.

$(x^2 + x^5)^2$
$= (x^2)^2 + 2(x^2)(x^5) + (x^5)^2$
$= x^4 + 2x^7 + x^{10}$

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The Notable Product