Question

Polynomials are everywhere. It is found in the slope of a hill, the curve of a bridge or the continuity of a mountain range.

Based on the given information, answer the following question.

If the hills are represented by the cubic polynomial $t\left(x\right)=p{x}^3+q{x}^2+rx+s$, then which of the following is always true?

• A. $s\ne 0$
• B. $r\ne 0$
• C. $q\ne 0$
• D. $p\ne 0$

Answer 1

The correct option is D $p\ne 0$

Given: Hills are represented by the cubic polynomial $t\left(x\right)=p{x}^3+q{x}^2+rx+s.$

If $s=0,\text{then}t\left(x\right)=p{x}^3+q{x}^2+rx$
$r=0,\text{then}t\left(x\right)=p{x}^3+q{x}^2+s$
$q=0,\text{then}t\left(x\right)=p{x}^3+rx+s$
All these are cubic polynomials since the highest degree is 3.

But when $p=0,\text{then}t\left(x\right)=q{x}^2+rx+s$ which is a quadratic equation.
Hence, $p$ cannot be equal to 0.