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?

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

Let us analyze the polynomial function:

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

Here, the coefficients are real numbers and the leading coefficient satisfies the condition:

$$a\ne 0$$

By definition, a value

$x=x_0$

is a root of the polynomial if and only if the polynomial evaluates to zero at that point, which means:

$$P\left(x_0\right)=0$$

To determine the condition under which the value

$x=0$

is a root of the polynomial, we substitute this value into the polynomial equation:

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

Simplifying the terms on the left-hand side, we get:

$$0+0+0+d=0$$

which simplifies directly to:

$$d=0$$

Since the terms containing the coefficients a, b, and c are all multiplied by zero when evaluating P(0), their values do not affect the result. Therefore, regardless of whatever real numbers are chosen for a, b, and c, we can always choose d = 0 to ensure that x = 0 is a root of the polynomial.