ax³ + bx² + cx + d is a polynomial on real x over real coefficients a, b, c, d wherein a ≠ 0. Which of the following statements is true?

Correct Answer: d can be chosen to ensure that x = 0 is a root for any given set a, b, c

Step-by-Step Explanation:

Let us analyze the given cubic polynomial equation with real coefficients:

$P\left(x\right)=a{x}^3+b{x}^2+cx+d$

A value $x=x_0$ is a root of the polynomial $P\left(x\right)$ if and only if $P\left(x_0\right)=0$.

We want $x=0$ to be a root of the polynomial. Evaluating $P\left(x\right)$ at $x=0$, we get:

$P\left(0\right)=a(0{)}^3+b(0{)}^2+c\left(0\right)+d$

$P\left(0\right)=d$

For $x=0$ to be a root of $P\left(x\right)$, we must have $P\left(0\right)=0$, which implies:

$d=0$

Regardless of what values are assigned to the coefficients $a$, $b$, and $c$ (with $a\ne 0$), simply setting the constant term $d=0$ guarantees that $P\left(0\right)=0$, thereby making $x=0$ a root of the polynomial.

Hence, the statement "d can be chosen to ensure that x = 0 is a root for any given set a, b, c" is strictly true.