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