Given the equation:
(x−1)⋅f(x)=3x4+x3−25x2+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:
Degree of f(x)=4−1=3
For the quadratic 2x2+mx+8 to have two distinct real roots, the discriminant must be positive. The discriminant Δ of a quadratic equation ax2+bx+c is given by:
Δ=b2−4ac
Substituting a=2, b=m, and c=8, we have:
Δ=m2−4⋅2⋅8=m2−64
For two distinct real roots, we require:
m2−64>0
Solving this inequality:
m2>64
Taking the square root of both sides, we find:
∣m∣>8
This implies:
m<−8orm>8
In interval notation, the possible values of m are:
(−∞,−8)∪(8,∞)
- The degree of f(x) is 3.
- The possible values of m are (−∞,−8)∪(8,∞).