Question

A polynomial $y = ax^3 + bx^2 + cx + d$ intersects the x-axis at 1 and -1, and the y-axis at 2. The value of $b$ is:

• A. $-2$
• B. $0$
• C. $1$
• D. $2$
• E. Cannot be determined

Hint:

Whenever polynomial roots are given, factorize using those roots, expand, and compare coefficients. The $y$-intercept condition helps fix the constants.

Answer 1

The Correct Option is A

Step 1: Factor from roots.
The polynomial intersects the x-axis at $x=1$ and $x=-1$. Hence, $(x-1)$ and $(x+1)$ are factors. Thus, \[ y = a(x-1)(x+1)(x-r), \] where $r$ is the third root. Step 2: Expand polynomial.
\[ y = a(x^2-1)(x-r) = a(x^3 - rx^2 - x + r). \] Comparing with $y = ax^3 + bx^2 + cx + d$: \[ a \text{Coefficient of $x^3$)}, b=-ar, c=-a, d=ar. \] Step 3: Use y-intercept.
At $x=0$, $y=d=2$. So, \[ d=ar=2. \hfill (1) \] Step 4: Eliminate $a$ using relation.
From coefficients: \[ b=-ar. \] But from (1), $ar=2$. So, \[ b=-2. \] \[ \boxed{-2} \]