Let's solve the given polynomials to determine their number of imaginary (complex) solutions.
### Part (a)
Polynomial: [tex]\( f(x) = x^4 - 7x^3 - x^2 + 67x - 60 \)[/tex]
To determine the number of imaginary solutions of this polynomial, we will use the fundamental theorem of algebra, which states that a polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots, counting multiplicities. These roots may be real or complex.
1. A polynomial of degree 4 ([tex]\( x^4 \)[/tex]) can have up to 4 roots.
2. By finding the exact roots, we can classify them as real or imaginary.
For quartic polynomials (degree 4), it's often easiest to find the roots numerically rather than trying to factorize analytically. Using numerical methods or root-finding algorithms (such as Newton's method), we can determine the nature of the roots (real or imaginary).
For this problem:
- After solving, we find that all the roots of the polynomial [tex]\( f(x) = x^4 - 7x^3 - x^2 + 67x - 60 \)[/tex] are real.
Thus, the number of imaginary roots in this case is [tex]\(0\)[/tex].
### Part (b)
Polynomial: [tex]\( g(x) = -4x + 4 \)[/tex]
This is a first-degree polynomial (linear equation).
1. A polynomial of degree 1 has exactly 1 root.
2. The root can be found by setting [tex]\( g(x) = 0 \)[/tex]:
[tex]\[
-4x + 4 = 0 \implies x = 1
\][/tex]
This is a real root.
Thus, the number of imaginary roots in this case is [tex]\(0\)[/tex].
### Part (c)
Polynomial: [tex]\( h(x) = x^2 + 2x - 3 \)[/tex]
To determine the nature of the roots of a quadratic polynomial, we can use the discriminant ([tex]\(\Delta\)[/tex]), which is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the coefficients of the polynomial [tex]\( ax^2 + bx + c \)[/tex].
For [tex]\( h(x) = x^2 + 2x - 3 \)[/tex]:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = 2\)[/tex]
- [tex]\(c = -3\)[/tex]
Calculating the discriminant:
[tex]\[
\Delta = b^2 - 4ac = 2^2 - 4(1)(-3) = 4 + 12 = 16
\][/tex]
Since the discriminant is positive ([tex]\(\Delta > 0\)[/tex]), the quadratic polynomial has two distinct real roots.
Thus, the number of imaginary roots in this case is [tex]\(0\)[/tex].
### Summary
- (a) [tex]\(x^4 - 7x^3 - x^2 + 67x - 60\)[/tex] has [tex]\(0\)[/tex] imaginary solutions.
- (b) [tex]\(-4x + 4\)[/tex] has [tex]\(0\)[/tex] imaginary solutions.
- (c) [tex]\(x^2 + 2x - 3\)[/tex] has [tex]\(0\)[/tex] imaginary solutions.
The answers are:
(a) 0
(b) 0
(c) 0
