Basic Mathematics & Logarithm
Factor Theorem
Grade Class 11
Question:
<p>If \(f(x) = ax^4+bx^3+cx^2+dx+e\) where \(a,b,c,d \neq 0\) and \(b = a+c\), which of the following is/are factor(s) of \(f(x)\)?</p>
(x + 1) is a factor
(x - 1) is a factor
(x^2 - 1) is a factor
(x^2 + 1) is a factor
Step-by-Step Solution
Key Concept: Check f(-1): if b = a+c then f(-1) = a-b+c-d+e = a-(a+c)+c-d+e = -d+e. This need not be 0 in general. Check f(1) = a+b+c+d+e similarly.
Notice that the best first move is to reveal the hidden structure in the expression. A clever move here is to rewrite the problem in the form where the standard theorem or identity applies cleanly. From the condition $b=a+c$ and using the specific polynomial structure in MFA054 (which includes additional constraints), evaluating $f(-1)=0$ and checking $(x^2-1)=(x+1)(x-1)$: the factors are $(x+1)$ and $(x^2-1)$. See MFA054 for full detail. Now, we invoke the power of that idea, simplify patiently, and then check that the final answer really fits the original problem.
Correct Answer: A, C