Question

Find the cube of 3x – 2y – 4z.
(a) \[27{{x}^{3}}-8{{y}^{3}}-64{{z}^{3}}-54{{x}^{2}}y+36x{{y}^{2}}-108{{x}^{2}}z-48{{y}^{2}}z+144x{{z}^{2}}-96y{{z}^{2}}+144xyz\]
(b) \[27{{x}^{3}}-8{{y}^{3}}-34{{z}^{3}}-54{{x}^{2}}y+36x{{y}^{2}}-108{{x}^{2}}z-48{{y}^{2}}z+144x{{z}^{2}}-96y{{z}^{2}}+144xyz\]
(c) \[27{{x}^{3}}-8{{y}^{3}}-24{{z}^{3}}-54{{x}^{2}}y+36x{{y}^{2}}-108{{x}^{2}}z-48{{y}^{2}}z+144x{{z}^{2}}-96y{{z}^{2}}+144xyz\]
(d) \[27{{x}^{3}}-8{{y}^{3}}-14{{z}^{3}}-54{{x}^{2}}y+36x{{y}^{2}}-108{{x}^{2}}z-48{{y}^{2}}z+144x{{z}^{2}}-96y{{z}^{2}}+144xyz\]

Answer 1

- Hint: First of all write the expression in the form [(3x) + (– 2y) + (– 4z)] and now cube it and use the formula \[{{\left( a+b+c \right)}^{3}}={{a}^{3}}+{{b}^{3}}+{{c}^{3}}+3\left( a+b \right)\left( b+c \right)\left( c+a \right)\] to get the cube of the given expression. Now, rearrange the terms and mark the correct answer.

Complete step-by-step solution -

In this question, we have to find the cube of 3x – 2y – 4z. Let us write the expression given in the question.
E = 3x – 2y – 4z
We have to find the cube of the above expression. So, by cubing both the sides of the above equation, we get,
\[M={{E}^{3}}={{\left( 3x-2y-4z \right)}^{3}}\]
We can also write the above expression as,
\[M={{\left[ \left( 3x \right)+\left( -2y \right)+\left( -4z \right) \right]}^{3}}\]
We know that
\[{{\left( a+b+c \right)}^{3}}={{a}^{3}}+{{b}^{3}}+{{c}^{3}}+3\left( a+b \right)\left( b+c \right)\left( c+a \right)\]
By using this in the above expression and considering a = 3x, b = – 2y and c = – 4z. We get,
\[M={{\left( 3x \right)}^{3}}+{{\left( -2y \right)}^{3}}+{{\left( -4z \right)}^{3}}+3\left( 3x-2y \right)\left( -2y-4z \right)\left( -4z+3x \right)\]
By simplifying the above equation, we get,
\[M=27{{x}^{3}}-8{{y}^{3}}-64{{z}^{3}}+3\left( 3x-2y \right)\left[ -2y\left( -4z+3x \right)-4z\left( -4z+3x \right) \right]\]
\[M=27{{x}^{3}}-8{{y}^{3}}-64{{z}^{3}}+\left( 9x-6y \right)\left[ 8yz-6xy+16{{z}^{2}}-12xz \right]\]
\[M=27{{x}^{3}}-8{{y}^{3}}-64{{z}^{3}}+9x\left( 8yz-6yx+16{{z}^{2}}-12xz \right)-6y\left( 8yz-6yx+16{{z}^{2}}-12xz \right)\]
\[M=27{{x}^{3}}-8{{y}^{3}}-64{{z}^{3}}+72xyz-54{{x}^{2}}y+144{{z}^{2}}x-108{{x}^{2}}z-48{{y}^{2}}z+36{{y}^{2}}x-96{{z}^{2}}y+72xyz\]
By grouping the like terms together in the above expression, we get,
\[M=\left( 27{{x}^{3}}-8{{y}^{3}}-64{{z}^{3}} \right)+\left( 72xyz+72xyz \right)-54{{x}^{2}}y+144{{z}^{2}}x-108{{x}^{2}}z-48{{y}^{2}}z+36{{y}^{2}}x-96{{z}^{2}}y\]
\[M=27{{x}^{3}}-8{{y}^{3}}-64{{z}^{3}}+144xyz-54{{x}^{2}}y+144{{z}^{2}}x-108{{x}^{2}}z-48{{y}^{2}}z+36{{y}^{2}}x-96{{z}^{2}}y\]
By rearranging the terms of the above equation, we get,
\[M=27{{x}^{3}}-8{{y}^{3}}-64{{z}^{3}}-54{{x}^{2}}y+36x{{y}^{2}}-108{{x}^{2}}z-48{{y}^{2}}z+144x{{z}^{2}}-96y{{z}^{2}}+144xyz\]
Hence, we get the cube of (3x – 2y – 4z) as \[27{{x}^{3}}-8{{y}^{3}}-64{{z}^{3}}-54{{x}^{2}}y+36x{{y}^{2}}-108{{x}^{2}}z-48{{y}^{2}}z+144x{{z}^{2}}-96y{{z}^{2}}+144xyz\]
So, option (a) is the right answer.

Note: In this question, students are advised to remember the formula of a cube of 3 terms that is \[{{\left( a+b+c \right)}^{3}}={{a}^{3}}+{{b}^{3}}+{{c}^{3}}+3\left( a+b \right)\left( b+c \right)\left( c+a \right)\]. But in case students forget the formula, they can still solve it by writing \[{{\left( a+b+c \right)}^{3}}\] as (a + b + c) (a + b + c) (a + b + c) and multiply it manually to get the correct answer though this method would be very lengthy and can even lead to mistakes. So, better is to remember the formula. Also, a, b and c must be properly taken with their signs.