Question

If $P\left(x\right)$ is a polynomial that has a simple zero at $x=4$ and a triple zero at $x=-4$, which of the following could be the factored form of $P\left(x\right)$ ?

• A. $P\left(x\right)=(x+4)(2x-8{)}^3$
• B. $P\left(x\right)=(x-4)(3x+12)$
• C. $P\left(x\right)=3(x+4)(2x-8)$
• D. $P\left(x\right)=(2x+8{)}^3(x-4)$

Answer 1

The correct option is D $P\left(x\right)=(2x+8{)}^3(x-4)$

The equation of the polynomial can be of the form $P\left(x\right)=(x+4{)}^n(x-4{)}^m$.

Now $x=-4$ is a triple zero, hence $n=3$ and $x=4$ is a simple zero, hence $m=1$.

Therefore, $P\left(x\right)=(x+4{)}^3(x-4{)}^1$ or $P\left(x\right)=K\left(\right(x+4{)}^3(x-4{)}^1$ where $k$ is a constant.
Option D can be represented in the above way.