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A fourth-degree polynomial is a polynomial of degree four. This means the highest power of the variable (usually x) is four. In general, a fourth-degree polynomial can be expressed as:
\[ ax^4 + bx^3 + cx^2 + dx + e \]
Here, a, b, c, d, and e are constants, and 'a' is not equal to zero. The given polynomial in the exercise is:
\[ f(x) = x^4 + 4x^3 + 6x^2 + 4x + 1 \]
Polynomials are used to represent various mathematical relationships and can have multiple roots or zeros. Identifying these roots is often crucial, as they provide valuable information about the behavior of the polynomial. For fourth-degree polynomials, you may encounter up to four roots.

Binomial expansion allows us to express a binomial raised to a power as a sum of terms. For example, the binomial theorem states:
\[ (a + b)^n = \binom{n}{0}a^n b^0 + \binom{n}{1}a^{n-1}b^1 + \binom{n}{2}a^{n-2}b^2 + \text{...} + \binom{n}{n}a^0 b^n \]
Here, \( \binom{n}{k} \) is a binomial coefficient. In our specific case, the polynomial \( f(x) = x^4 + 4x^3 + 6x^2 + 4x + 1 \) resembles the binomial expansion of \( (x+1)^4 \). We can expand \( (x+1)^4 \) by applying the binomial theorem:
\( (x+1)^4 = \binom{4}{0}x^4 + \binom{4}{1}x^3 \times 1 + \binom{4}{2}x^2 \times 1^2 + \binom{4}{3}x \times 1^3 + \binom{4}{4}1^4 \)
This simplifies to \( x^4 + 4x^3 + 6x^2 + 4x + 1 \), confirming that our polynomial can be factored into: \( (x+1)^4 \).

Repeated roots occur when a polynomial has the same root multiple times. This is significant because it affects the shape of the graph and the behavior of the polynomial function. For the polynomial \( (x+1)^4 \), setting it equal to zero, we solve:
\[ (x+1)^4 = 0 \]
Taking the fourth root of both sides, we get:
\[ x+1 = 0 \]
Which simplifies to:
\[ x = -1 \]
Since we started with \( (x+1)^4 \), the root \( x = -1 \) is repeated four times. Therefore, the polynomial's roots are \( x = -1, -1, -1, -1 \). Recognizing repeated roots is crucial as they indicate the multiplicity of the root which impacts the behavior of the polynomial graph especially around these points.