Questions: Suppose that f is a polynomial such that (x-1) * f(x) = 3x^4 + x^3 - 25x^2 + 38x - 17 What is the degree of f? m is a real number and 2x^2 + mx + 8 has two distinct real roots, then what are the possible values of m? Express your answer in interval notation.

Solution Steps

Given the equation:

\[ (x-1) \cdot f(x) = 3x^4 + x^3 - 25x^2 + 38x - 17 \]

The polynomial on the right-hand side is of degree 4. Since \((x-1)\) is a polynomial of degree 1, the degree of \( f(x) \) must be:

\[ \text{Degree of } f(x) = 4 - 1 = 3 \]

For the quadratic \( 2x^2 + mx + 8 \) to have two distinct real roots, the discriminant must be positive. The discriminant \(\Delta\) of a quadratic equation \( ax^2 + bx + c \) is given by:

\[ \Delta = b^2 - 4ac \]

Substituting \( a = 2 \), \( b = m \), and \( c = 8 \), we have:

\[ \Delta = m^2 - 4 \cdot 2 \cdot 8 = m^2 - 64 \]

For two distinct real roots, we require:

\[ m^2 - 64 > 0 \]

Solving this inequality:

\[ m^2 > 64 \]

Taking the square root of both sides, we find:

\[ |m| > 8 \]

This implies:

\[ m < -8 \quad \text{or} \quad m > 8 \]

In interval notation, the possible values of \( m \) are:

\[ (-\infty, -8) \cup (8, \infty) \]

Final Answer