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Question
Simplify:
(5 − x)(6 − 5x)( 2 − x)
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Solution
To simplify, we will proceed as follows:
\[\left( 5 - x \right)\left( 6 - 5x \right)\left( 2 - x \right)\]
\[ = \left[ \left( 5 - x \right)\left( 6 - 5x \right) \right]\left( 2 - x \right)\]
\[= \left[ 5\left( 6 - 5x \right) - x\left( 6 - 5x \right) \right]\left( 2 - x \right)\] (Distributive law)
\[= \left( 30 - 25x - 6x + 5 x^2 \right)\left( 2 - x \right)\]
\[ = \left( 30 - 31x + 5 x^2 \right)\left( 2 - x \right)\]
\[ = 2\left( 30 - 31x + 5 x^2 \right) - x\left( 30 - 31x + 5 x^2 \right)\]
\[ = 60 - 62x + 10 x^2 - 30x + 31 x^2 - 5 x^3\]
\[= 60 - 62x - 30x + 10 x^2 + 31 x^2 - 5 x^3\] (Rearranging)
\[= 60 - 92x + 41 x^2 - 5 x^3\] (Combining like terms)
Thus, the answer is \[60 - 92x + 41 x^2 - 5 x^3\].
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