Step 1

Consider the given equation.

\(\displaystyle{x}^{{{3}}}-{8}={x}-{2}\)

Step 2

On solving above equation,

\(\displaystyle{x}^{{{3}}}-{8}-{x}+{2}={0}\)

\(\displaystyle{x}^{{{3}}}-{x}-{6}={0}\)

Step 3

Since, the one factor of this equation is (x – 2), so

\(\displaystyle\Rightarrow{\frac{{{x}^{{{3}}}-{x}-{6}}}{{{x}-{2}}}}\)

\(\displaystyle\Rightarrow{x}^{{{2}}}+{2}{x}+{3}\)

Step 4

Now, solving the above quadratic equation

\(\displaystyle{x}^{{{2}}}+{2}{x}+{3}={0}\)

Step 5

Formula used:

\(\displaystyle{x}={\frac{{-{b}\pm\sqrt{{{b}^{{{2}}}-{4}{a}{c}}}}}{{{2}{a}}}}\)

Step 6

Therefore,

\(\displaystyle{x}={\frac{{-{2}\pm\sqrt{{{2}^{{{2}}}-{4}\times{1}\times{3}}}}}{{{2}\times{1}}}}\)

\(\displaystyle{x}={\frac{{-{2}\pm\sqrt{{{4}-{12}}}}}{{{2}}}}\)

\(\displaystyle{x}={\frac{{-{2}\pm\sqrt{{-{8}}}}}{{{2}}}}\)

\(\displaystyle{x}={\frac{{-{2}\pm{2}{i}\sqrt{{{2}}}}}{{{2}}}}\)

\(\displaystyle{x}=-{1}\pm{i}\sqrt{{{2}}}\)

Step 7

Answer: The factors of the equation is equal to

\(\displaystyle{2},-{1}+{i}\sqrt{{{2}}},-{1}-{i}\sqrt{{{2}}}\)

Consider the given equation.

\(\displaystyle{x}^{{{3}}}-{8}={x}-{2}\)

Step 2

On solving above equation,

\(\displaystyle{x}^{{{3}}}-{8}-{x}+{2}={0}\)

\(\displaystyle{x}^{{{3}}}-{x}-{6}={0}\)

Step 3

Since, the one factor of this equation is (x – 2), so

\(\displaystyle\Rightarrow{\frac{{{x}^{{{3}}}-{x}-{6}}}{{{x}-{2}}}}\)

\(\displaystyle\Rightarrow{x}^{{{2}}}+{2}{x}+{3}\)

Step 4

Now, solving the above quadratic equation

\(\displaystyle{x}^{{{2}}}+{2}{x}+{3}={0}\)

Step 5

Formula used:

\(\displaystyle{x}={\frac{{-{b}\pm\sqrt{{{b}^{{{2}}}-{4}{a}{c}}}}}{{{2}{a}}}}\)

Step 6

Therefore,

\(\displaystyle{x}={\frac{{-{2}\pm\sqrt{{{2}^{{{2}}}-{4}\times{1}\times{3}}}}}{{{2}\times{1}}}}\)

\(\displaystyle{x}={\frac{{-{2}\pm\sqrt{{{4}-{12}}}}}{{{2}}}}\)

\(\displaystyle{x}={\frac{{-{2}\pm\sqrt{{-{8}}}}}{{{2}}}}\)

\(\displaystyle{x}={\frac{{-{2}\pm{2}{i}\sqrt{{{2}}}}}{{{2}}}}\)

\(\displaystyle{x}=-{1}\pm{i}\sqrt{{{2}}}\)

Step 7

Answer: The factors of the equation is equal to

\(\displaystyle{2},-{1}+{i}\sqrt{{{2}}},-{1}-{i}\sqrt{{{2}}}\)