Question

Choose the correct option for the following statement.

$x+1$ and $2x-3$ are factors of $2x^3-9x^2+x+12.$

• A. True
• B. False
• C. Ambiguous
• D. Data insufficient

Answer 1

The correct option is A True

To prove that $(x+1)$ and $(2x-3)$ are factors of $p\left(x\right)=2x^3-9x^2+x+12$ it is sufficient to show that p(-1) and $p\left(\frac{3}{2}\right)$ both are equal to zero.

$p(-1)=2(-1{)}^3-9(-1{)}^2+(-1)+12=-2-9-1+12$
$=-12+12=0$

And $p\left(\frac{3}{2}\right)=2{\left(\frac{3}{2}\right)}^3-9{\left(\frac{3}{2}\right)}^2+\left(\frac{3}{2}\right)+12$

$=\frac{27}{4}-\frac{81}{4}+\frac{3}{2}+12=\frac{27-81+6+48}{4}=\frac{-81+81}{4}=0$

$x+1$ and $2x-3$ are factors of $2x^3-9x^2+x+12$ is true.