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.

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.
Transcript text: Suppose that $f$ is a polynomial such that \[ (x-1) \cdot f(x)=3 x^{4}+x^{3}-25 x^{2}+38 x-17 \] What is the degree of $f$ ? $m$ is a real number and $2 x^{2}+m x+8$ has two distinct real roots, then what are the possible values of $m$ ? Express your answer in interval nolation.
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Solution

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Solution Steps

Step 1: Determine the Degree of f(x) f(x)

Given the equation:

(x1)f(x)=3x4+x325x2+38x17 (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 (x1)(x-1) is a polynomial of degree 1, the degree of f(x) f(x) must be:

Degree of f(x)=41=3 \text{Degree of } f(x) = 4 - 1 = 3

Step 2: Determine the Possible Values of m m

For the quadratic 2x2+mx+8 2x^2 + mx + 8 to have two distinct real roots, the discriminant must be positive. The discriminant Δ\Delta of a quadratic equation ax2+bx+c ax^2 + bx + c is given by:

Δ=b24ac \Delta = b^2 - 4ac

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

Δ=m2428=m264 \Delta = m^2 - 4 \cdot 2 \cdot 8 = m^2 - 64

For two distinct real roots, we require:

m264>0 m^2 - 64 > 0

Solving this inequality:

m2>64 m^2 > 64

Taking the square root of both sides, we find:

m>8 |m| > 8

This implies:

m<8orm>8 m < -8 \quad \text{or} \quad m > 8

In interval notation, the possible values of m m are:

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

Final Answer

  • The degree of f(x) f(x) is 3\boxed{3}.
  • The possible values of m m are (,8)(8,)\boxed{(-\infty, -8) \cup (8, \infty)}.
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