Question

The value of $(x-1)(x+\frac{1}{2}-\frac{\sqrt{3}}{2}i)(x+\frac{1}{2}+\frac{\sqrt{3}}{2}i)$ is

• A. $x^3+x^2+x1$
• B. $x^3-1$
• C. $x^3+1$
• D. $x^3-x^2+x+1$

Answer 1

Answer: (B)

$x^3-1$

$(x-1)(x+\frac{1}{2}-\frac{\sqrt{3}}{2}i)(x+\frac{1}{2}+\frac{\sqrt{3}}{2}i)$

$=(x-1)[(x+\frac{1}{2})-\left(\frac{\sqrt{3}}{2}i\right)][(x+\frac{1}{2})+\left(\frac{\sqrt{3}}{2}i\right)]$

$=(x-1)[{(x+\frac{1}{2})}^2-{\left(\frac{\sqrt{3}}{2}i\right)}^2]$

$=(x-1)[x^2+x+\frac{1}{4}+\frac{3}{4}]$

$=(x-1)[x^2+x+1]$

$=x^3+x^2+x-x^2-x-1$

$=x^3-1$

Hence, the answer is $x^3-1.$