Certainly! Let's analyze each polynomial equation step-by-step and determine the number of possible imaginary (complex) solutions.
### Part (a)
Consider the polynomial equation:
[tex]\[ x^4 - 7x^3 - x^2 + 67x - 60 = 0 \][/tex]
The highest degree of the polynomial is 4, indicating that there are a total of 4 solutions, real or complex. For this polynomial, all 4 roots have been verified to be real numbers. Hence, the number of imaginary (complex) solutions is:
[tex]\[ 0 \][/tex]
### Part (b)
Consider the polynomial equation:
[tex]\[ -4x + 4 = 0 \][/tex]
This is a linear equation (degree 1). The solution to this equation is:
[tex]\[ x = 1 \][/tex]
Since this is a real number, there are no complex (imaginary) solutions. Thus, the number of imaginary solutions is:
[tex]\[ 0 \][/tex]
### Part (c)
Consider the quadratic equation:
[tex]\[ x^2 + 2x - 3 = 0 \][/tex]
The solutions to a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] are given by the quadratic formula:
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = 2\)[/tex], [tex]\(c = -3\)[/tex]. Let's calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac = 2^2 - 4(1)(-3) = 4 + 12 = 16 \][/tex]
Since the discriminant is positive ([tex]\(16 > 0\)[/tex]), there are 2 distinct real roots. Thus, there are no complex solutions. Therefore, the number of imaginary (complex) solutions is:
[tex]\[ 0 \][/tex]
To summarize, the number of imaginary (complex) solutions for each part is as follows:
[tex]\[ a) 0 \][/tex]
[tex]\[ b) 0 \][/tex]
[tex]\[ c) 0 \][/tex]
