The least degree of a polynomial with integer coefficient whose one of the roots may be cos `12^@` is

Answer

To find the least degree of a polynomial with integer coefficients whose one of the roots is \( \cos 12^\circ \), we can follow these steps:

### Step 1: Identify the angle and its cosine
We are given that one of the roots is \( \cos 12^\circ \).

### Step 2: Use multiple angle formulas
We can utilize the multiple angle formula for cosine. Specifically, we can express \( \cos 5x \) in terms of \( \cos x \):
\[ \cos 5x = 16 \cos^5 x - 20 \cos^3 x + 5 \cos x \]
Setting \( x = 12^\circ \), we have:
\[ \cos 60^\circ = \frac{1}{2} \]
Thus, we can write:
\[ \cos 60^\circ = 16 \cos^5 12^\circ - 20 \cos^3 12^\circ + 5 \cos 12^\circ \]

### Step 3: Substitute \( \cos 12^\circ \) with \( x \)
Let \( x = \cos 12^\circ \). Then the equation becomes:
\[ \frac{1}{2} = 16x^5 - 20x^3 + 5x \]

### Step 4: Rearranging the equation
Rearranging gives:
\[ 16x^5 - 20x^3 + 5x - \frac{1}{2} = 0 \]
To eliminate the fraction, multiply through by 2:
\[ 32x^5 - 40x^3 + 10x - 1 = 0 \]

### Step 5: Identify the polynomial
The polynomial we have is:
\[ f(x) = 32x^5 - 40x^3 + 10x - 1 \]

### Step 6: Determine the degree of the polynomial
The degree of this polynomial is 5, since the highest power of \( x \) is 5.

### Step 7: Check if it can be factored further
To determine if there is a polynomial of lower degree with integer coefficients that has \( \cos 12^\circ \) as a root, we need to check if \( f(x) \) can be factored. However, since \( \cos 12^\circ \) is not a rational number, and the polynomial is irreducible over the integers, we conclude that the least degree of a polynomial with integer coefficients whose one of the roots is \( \cos 12^\circ \) is indeed 5.

### Final Answer
The least degree of the polynomial is **5**.

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