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Understanding polynomial roots is essential for solving many algebraic problems. Essentially, a root of a polynomial is any value for which the polynomial equation equals zero. For instance, if we have a polynomial like \( P(x) = (x - r)Q(x) \), then \( r \) is called a root of the polynomial, because plugging \( r \) into \( P(x) \) yields zero: \( P(r) = 0 \). In the given exercise, the specified roots are 5 and 3-2i, which indicate the values at which the polynomial will intersect or touch the x-axis on a graph.

In practical terms, each root corresponds to a factor in the polynomial. The polynomial with roots 5, 3-2i, and its conjugate 3+2i can be constructed by setting up factors that become zero at these roots: \( (x - 5)(x - (3 - 2i))(x - (3 + 2i)) \). A noteworthy point is the fundamental theorem of algebra, which states that a polynomial of degree \( n \) will have exactly \( n \) roots, although some may be repeated or complex, as in the case outlined above.

When dealing with polynomials with real coefficients, the complex roots will always come in pairs, known as complex conjugate pairs. This concept stems from the fact that if a complex number is a root, its conjugate is also a root. A complex conjugate of a complex number involves changing the sign between the two parts of the number; for \( 3-2i \), the conjugate is \( 3+2i \).

In the context of the provided problem, since one of the given roots is the complex number \( 3-2i \), its conjugate \( 3+2i \) must also be a root. This creates symmetry in the complex plane and ensures that the resultant polynomial has real coefficients. Multiplying these conjugate pairs together, as shown in Step 3 of the solution, generates a quadratic polynomial with real coefficients, because the product of two complex conjugates is always real. For example, multiplying the pair yields \( (x-(3-2i))(x-(3+2i)) = (x-3)^2 - (2i)^2 \), which simplifies to a polynomial with real coefficients.

Polynomial coefficients are the numbers that multiply the variables that are raised to powers in polynomial expressions. These coefficients provide much information about the polynomial, such as its degree and its behavior as the input values become very large or very small.

For the polynomial in question, arrived at in Step 4, the coefficients are real numbers that precede the terms \( x^3 \) (which is 1), \( x^2 \) (which is -11), \( x \) (which is 49), and the constant term (which is -105). Coefficients play a critical role in determining the shape and the symmetry of the polynomial's graph. They also affect the end behavior, such as how the polynomial will behave towards infinity or negative infinity. Importantly, these coefficients must satisfy certain relationships if the polynomial's roots are known. In algebraic courses, students explore these relationships through equations like Vieta's formulas, which connect the roots of polynomials to their coefficients.