Question

The total number of polynomials of the form \[ x^{3}+ax^{2}+bx+c \] which are divisible by \(x^{2}+1\), where \(a,b,c\in\{1,2,3,\dots,10\}\) is:

• A. 120
• B. 45
• C. 10
• D. 15

Hint:

You can also solve this using polynomial long division! Dividing $x^3+ax^2+bx+c$ by $x^2+1$ leaves a remainder of $(b-1)x + (c-a)$. For the polynomial to be perfectly divisible, this remainder must be zero, which instantly gives you the equations $b=1$ and $c=a$.

Answer 1

The Correct Option is C

Concept: For a polynomial $P(x)$ to be perfectly divisible by a quadratic factor $(x^2+1)$, the complex roots of that factor ($x = \pm i$) must also be roots of the polynomial itself. This condition gives us a system of equations by setting the real and imaginary parts of $P(i) = 0$ to zero. Step 1: Substitute the root $x = i$ into the polynomial.
Let our polynomial function be $P(x) = x^3 + ax^2 + bx + c$. Since it is divisible by $x^2 + 1$: $$P(i) = 0 \quad \Rightarrow \quad (i)^3 + a(i)^2 + b(i) + c = 0$$

Step 2: Separate the real and imaginary parts.
Recall the powers of the imaginary unit: $i^2 = -1$ and $i^3 = -i$. Substitute these into our equation: $$-i - a + bi + c = 0$$ Group the real terms and the imaginary terms together: $$(c - a) + i(b - 1) = 0$$

Step 3: Solve for the coefficient constraints.
For a complex number to equal zero, its real part and its imaginary part must both equal zero independently:
• $\text{Real part: } c - a = 0 \implies c = a$
• $\text{Imaginary part: } b - 1 = 0 \implies b = 1$

Step 4: Count the total number of valid configurations.
We are given that the coefficients must be chosen from the set of integers from 1 to 10: $\{1, 2, 3, \dots, 10\}$. Let us count the number of valid choices for each coefficient based on our constraints:
• The coefficient $b$ is uniquely fixed to a single value: $b = 1$ (1 choice).
• The coefficient $a$ can be chosen freely from any of the 10 available integers in the set (10 choices).
• Once $a$ is chosen, the coefficient $c$ is uniquely fixed because it must match $a$ ($c = a$) (1 choice). Using the fundamental counting principle, the total number of unique polynomials that can be formed is: $$\text{Total Polynomials} = 10 \times 1 \times 1 = 10$$ This matches option (C) perfectly.