Polynomials
Factoring x⁴ - nx + 63 into linear × cubic
MJAT_TS8_P2
Grade 12
Question:
If $n$ is the smallest positive integer such that $x^4-nx+63$ can be written as a product of a linear and a cubic polynomial with integer coefficients, then:
A) Number of divisors of $n$ is 10
B) Number of solutions of $ab=n$ ($a,b\in\mathbb{N}$) is 10
C) Number of odd divisors of $n$ is 2
D) $\displaystyle\int_0^n e^{x[x]}\,dx=100(e-1)$
Step-by-Step Solution
Key Concept: $(x-\alpha)(x^3+ax^2+bx+c)=x^4-nx+63$. Expanding: $-\alpha=-0$ (no $x^3$ term)... actually $a=\alpha$, $b=a\alpha$, $c=b\alpha$, $-c\alpha=63$. So $\alpha^4=63$? No: from solution $n=\alpha^3+63/\alpha$ is minimized at $\alpha=3$ giving $n=27+21=48$.
A ✓, B ✓, C ✓. Answer: A, B, C.
Correct Answer: ABC