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How do you identify special products when factoring?

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Last updated date: 18th Jun 2024
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Answer
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Hint: Special products is an often-used mathematical term under which factors are combined in order to form products. They are called ‘special’ as they don’t require any long solutions. There are four types of special products, which are,
-Square of a binomial- This type of special product results in a perfect square trinomial.
-Product of sum and difference of the two binomials- This type of special product results in the difference of two squares.
-Square of trinomials- This type of special product results in six terms.
-Product of binomials- This type of special product results in the general trinomial.

Complete step by step solution:
Here, in this question we are asked how to identify special products when factoring.
The most used and common method or way to identify special products while factorising is by square roots if it is a perfect square or a perfect cube. A perfect square is a number which can be expressed as the square of a number. For example-$25$ is a perfect square because ${\left( 5 \right)^2} = 25$.

A perfect cube is a number which can be expressed as the cube of a number. For example-$27$ is a perfect cube because ${\left( 3 \right)^3} = 27$. An example of a special product is the difference of two squares:
${x^2} - {y^2} = \left( {x + y} \right)\left( {x - y} \right)$
$
\Rightarrow {\left( {x + y} \right)^2} = {x^2} + {y^2} + 2xy \\
\therefore{\left( {x - y} \right)^2} = {x^2} + {y^2} - 2xy \\ $
It is important to note that, ${\left( {x - y} \right)^2} \ne {x^2} - {y^2}$.

Note: Students should note that$\left( {x + a} \right)\left( {x + b} \right) = x\left( {x + b} \right) + a\left( {x + b} \right)$ (by distributivity)
$ \Rightarrow {x^2} + xb + ax + ab$
$ \Rightarrow {x^2} + bx + ax + ab$ (since, $xb = bx$)
$ \Rightarrow {x^2} + ax + bx + ab$ (since, $bx + ax = ax + bx$)
$ \Rightarrow {x^2} + \left( {a + b} \right)x + ab$
Therefore, we have the following identity$\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab$. Further, by using this identity we can have various results.