Question

The expression $27{{x}^{3}}+8{{y}^{3}}$ can be expressed in factors as
a. ${{\left( 3x-2y \right)}^{3}}$
b. ${{\left( 3x+2y \right)}^{3}}$
c. ${{\left( 2x+3y \right)}^{3}}$
d. None of these.

Answer 1

Hint: We will first write the given expression, that is, $27{{x}^{3}}+8{{y}^{3}}$ as ${{\left( 3x \right)}^{3}}+{{\left( 2y \right)}^{3}}$ and then we will make use of the formula, ${{x}^{3}}+{{y}^{3}}=\left( x+y \right)\left[ {{x}^{2}}+{{y}^{2}}-xy \right]$ in order to express $27{{x}^{3}}+8{{y}^{3}}$ in terms of its factors.

Complete step-by-step answer:
It is given in the question that we have to express $27{{x}^{3}}+8{{y}^{3}}$ in terms of its factors. Now, we know that 27 can be expressed as $3\times 3\times 3$ and 8 can be expressed as $2\times 2\times 2$. Therefore, we can write the expression $27{{x}^{3}}+8{{y}^{3}}$ as ${{\left( 3x \right)}^{3}}+{{\left( 2y \right)}^{3}}$.
Now, we know the identity that ${{x}^{3}}+{{y}^{3}}=\left( x+y \right)\left[ {{x}^{2}}+{{y}^{2}}-xy \right]$. So, on applying this identity we can write ${{\left( 3x \right)}^{3}}+{{\left( 2y \right)}^{3}}$ as follows,
${{\left( 3x \right)}^{3}}+{{\left( 2y \right)}^{3}}=\left( 3x+2y \right)\left[ {{\left( 3x \right)}^{2}}+{{\left( 2y \right)}^{2}}-\left( 3x \right)\left( 2y \right) \right]$
We can simplify the above expression and further write it as shown below,
${{\left( 3x \right)}^{3}}+{{\left( 2y \right)}^{3}}=\left( 3x+2y \right)\left[ 9{{x}^{2}}+4{{y}^{2}}-6xy \right]$
Now, if we compare our obtained result with the options given in the question, then we can see that it does not match any of the given options, that is it does not match options (a), (b) or(c).
Hence, we can say that option (d) is the correct answer.

Note: Most of the students make mistake while using the identity and they write it as ${{x}^{3}}+{{y}^{3}}=\left( x-y \right)\left[ {{x}^{2}}+{{y}^{2}}+xy \right]$, but this is wrong and it will result in the formation of wrong answer. So, the students must be very careful about the signs used inside the identity. Hence, the students should note that the correct formula is ${{x}^{3}}+{{y}^{3}}=\left( x+y \right)\left[ {{x}^{2}}+{{y}^{2}}-xy \right]$, in order to avoid making any mistakes.