Question
Simplify the following: $(x + 3)^3 + (x - 3)^3$
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Answer
In the given problem, we have to simplify equation Given $(x + 3)^3 + (x - 3)^3$
We shall use the identity $a^3 + b^3 = (a + b)(a^2 + b^2 - ab)$
Here $a= (x + 3), b = (x - 3)$ By applying identity
we get $=\big(\text{x}+\not3+\text{x}-\not3\big)\big[(\text{x}+3)^2+(\text{x}-3)^2-(\text{x}+3)(\text{x}-3)\big]$
$=2\text{x}\big[\big(\text{x}^2+3^2+2\times\text{x}\times3\big)+\big(\text{x}^2+3^2-2\times\text{x}\times3\big)-(\text{x}^2)-3^2\big]$
$=2\text{x}\big[\big(\text{x}^2+9+6\text{x}\big)+\big(\text{x}^2+9-6\text{x}\big)-\big(\text{x}^2-3^2\big)\big]$
$=2\text{x}\big[\text{x}^2+9+6\text{x}+\text{x}^2+9-6\text{x}-\text{x}^2+9\big]$
$=2\text{x}\big[\text{x}^2+\not\text{x}^2-\not\text{x}^2-6\text{x}+6\text{x}+9+9+9\big]$
$=2\text{x}\big[\text{x}^2+27\big]$
$=2\text{x}^2+54\text{x}$
Hence simplified form of expression $(\text{x}+3)^3+(\text{x}-3)^3$ is
$2\text{x}^2+54\text{x}.$
We shall use the identity $a^3 + b^3 = (a + b)(a^2 + b^2 - ab)$
Here $a= (x + 3), b = (x - 3)$ By applying identity
we get $=\big(\text{x}+\not3+\text{x}-\not3\big)\big[(\text{x}+3)^2+(\text{x}-3)^2-(\text{x}+3)(\text{x}-3)\big]$
$=2\text{x}\big[\big(\text{x}^2+3^2+2\times\text{x}\times3\big)+\big(\text{x}^2+3^2-2\times\text{x}\times3\big)-(\text{x}^2)-3^2\big]$
$=2\text{x}\big[\big(\text{x}^2+9+6\text{x}\big)+\big(\text{x}^2+9-6\text{x}\big)-\big(\text{x}^2-3^2\big)\big]$
$=2\text{x}\big[\text{x}^2+9+6\text{x}+\text{x}^2+9-6\text{x}-\text{x}^2+9\big]$
$=2\text{x}\big[\text{x}^2+\not\text{x}^2-\not\text{x}^2-6\text{x}+6\text{x}+9+9+9\big]$
$=2\text{x}\big[\text{x}^2+27\big]$
$=2\text{x}^2+54\text{x}$
Hence simplified form of expression $(\text{x}+3)^3+(\text{x}-3)^3$ is
$2\text{x}^2+54\text{x}.$
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