Question:
Let \(a_n,a_{n-1},\ldots,a_1,a_0\in \mathbb{C}\) and
\[
f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0
\]
is a polynomial. If the polynomial \(f(x)\) is monic, then
Let \(a_n,a_{n-1},\ldots,a_1,a_0\in \mathbb{C}\) and
\[
f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0
\]
is a polynomial. If the polynomial \(f(x)\) is monic, then
Show Hint
A monic polynomial is a polynomial whose leading coefficient is exactly \(1\).
Updated On: Jun 25, 2026
- \(a_n\neq 0\)
- \(a_n=1\)
- \(a_n\gt 0\)
- \(a_n\lt 0\)
Show Solution
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The Correct Option is B
Solution and Explanation
Step 1: Understand the meaning of a monic polynomial.
A polynomial is called monic if its leading coefficient is equal to \(1\).
Step 2: Identify the leading term.
The given polynomial is \[ f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 \] Here, the highest power of \(x\) is \[ x^n \] So the leading term is \[ a_nx^n \] Therefore, the leading coefficient is \[ a_n \]
Step 3: Apply the condition for a monic polynomial.
Since \(f(x)\) is monic, its leading coefficient must be \(1\).
Hence, \[ a_n=1 \]
Step 4: Final conclusion.
Therefore, \[ \boxed{a_n=1} \]
A polynomial is called monic if its leading coefficient is equal to \(1\).
Step 2: Identify the leading term.
The given polynomial is \[ f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 \] Here, the highest power of \(x\) is \[ x^n \] So the leading term is \[ a_nx^n \] Therefore, the leading coefficient is \[ a_n \]
Step 3: Apply the condition for a monic polynomial.
Since \(f(x)\) is monic, its leading coefficient must be \(1\).
Hence, \[ a_n=1 \]
Step 4: Final conclusion.
Therefore, \[ \boxed{a_n=1} \]
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