Answer :
To find the zeros of the polynomial [tex]\( f(x) = x^3 - 5x^2 - 3x - 18 \)[/tex], follow these detailed steps:
### Step 1: Understand the Polynomial Function
We have a cubic polynomial, [tex]\( f(x) = x^3 - 5x^2 - 3x - 18 \)[/tex]. A cubic polynomial can have up to three zeros, which are the solutions to the equation [tex]\( f(x) = 0 \)[/tex].
### Step 2: Solve for the Zeros
The zeros of the polynomial are the values of [tex]\( x \)[/tex] that satisfy [tex]\( f(x) = 0 \)[/tex].
The zeros for this polynomial are found to be:
- [tex]\( x = 6 \)[/tex]
- [tex]\( x = -\frac{1}{2} - \frac{\sqrt{11}i}{2} \)[/tex]
- [tex]\( x = -\frac{1}{2} + \frac{\sqrt{11}i}{2} \)[/tex]
### Step 3: Interpret the Zeros
1. Real Zero:
- [tex]\( x_1 = 6 \)[/tex]: This is a real zero. It is the value at which the function crosses the x-axis.
2. Complex Conjugate Zeros:
- [tex]\( x_2 = -\frac{1}{2} - \frac{\sqrt{11}i}{2} \)[/tex]
- [tex]\( x_3 = -\frac{1}{2} + \frac{\sqrt{11}i}{2} \)[/tex]
These are complex zeros and they occur in conjugate pairs (since the polynomial has real coefficients). They imply that the polynomial does not cross the x-axis at these points but instead gives rise to oscillations or rotations in the complex plane.
### Step 4: Relate Zeros to the Graph
1. Real Zero at [tex]\( x = 6 \)[/tex]:
- At [tex]\( x = 6 \)[/tex], the graph of the polynomial crosses or touches the x-axis. Since [tex]\( x = 6 \)[/tex] is a single zero, we can infer that the graph will cross the x-axis at this point (rather than just touching it without crossing).
2. Complex Zeros:
- The zeros [tex]\( x = -\frac{1}{2} - \frac{\sqrt{11}i}{2} \)[/tex] and [tex]\( x = -\frac{1}{2} + \frac{\sqrt{11}i}{2} \)[/tex] do not correspond to x-axis crossings, but their influence is observed in the shape of the graph. The graph will exhibit a certain smooth curvature, demonstrating the polynomial’s roots structure in the complex plane.
### Conclusion
The polynomial [tex]\( f(x) = x^3 - 5x^2 - 3x - 18 \)[/tex] has one real zero [tex]\( x = 6 \)[/tex] and two complex conjugate zeros [tex]\( x = -\frac{1}{2} - \frac{\sqrt{11}i}{2} \)[/tex] and [tex]\( x = -\frac{1}{2} + \frac{\sqrt{11}i}{2} \)[/tex]. The real zero indicates where the function intersects the x-axis, while the complex zeros affect the curvature and shape of the graph but do not show up as x-axis intersections.
### Step 1: Understand the Polynomial Function
We have a cubic polynomial, [tex]\( f(x) = x^3 - 5x^2 - 3x - 18 \)[/tex]. A cubic polynomial can have up to three zeros, which are the solutions to the equation [tex]\( f(x) = 0 \)[/tex].
### Step 2: Solve for the Zeros
The zeros of the polynomial are the values of [tex]\( x \)[/tex] that satisfy [tex]\( f(x) = 0 \)[/tex].
The zeros for this polynomial are found to be:
- [tex]\( x = 6 \)[/tex]
- [tex]\( x = -\frac{1}{2} - \frac{\sqrt{11}i}{2} \)[/tex]
- [tex]\( x = -\frac{1}{2} + \frac{\sqrt{11}i}{2} \)[/tex]
### Step 3: Interpret the Zeros
1. Real Zero:
- [tex]\( x_1 = 6 \)[/tex]: This is a real zero. It is the value at which the function crosses the x-axis.
2. Complex Conjugate Zeros:
- [tex]\( x_2 = -\frac{1}{2} - \frac{\sqrt{11}i}{2} \)[/tex]
- [tex]\( x_3 = -\frac{1}{2} + \frac{\sqrt{11}i}{2} \)[/tex]
These are complex zeros and they occur in conjugate pairs (since the polynomial has real coefficients). They imply that the polynomial does not cross the x-axis at these points but instead gives rise to oscillations or rotations in the complex plane.
### Step 4: Relate Zeros to the Graph
1. Real Zero at [tex]\( x = 6 \)[/tex]:
- At [tex]\( x = 6 \)[/tex], the graph of the polynomial crosses or touches the x-axis. Since [tex]\( x = 6 \)[/tex] is a single zero, we can infer that the graph will cross the x-axis at this point (rather than just touching it without crossing).
2. Complex Zeros:
- The zeros [tex]\( x = -\frac{1}{2} - \frac{\sqrt{11}i}{2} \)[/tex] and [tex]\( x = -\frac{1}{2} + \frac{\sqrt{11}i}{2} \)[/tex] do not correspond to x-axis crossings, but their influence is observed in the shape of the graph. The graph will exhibit a certain smooth curvature, demonstrating the polynomial’s roots structure in the complex plane.
### Conclusion
The polynomial [tex]\( f(x) = x^3 - 5x^2 - 3x - 18 \)[/tex] has one real zero [tex]\( x = 6 \)[/tex] and two complex conjugate zeros [tex]\( x = -\frac{1}{2} - \frac{\sqrt{11}i}{2} \)[/tex] and [tex]\( x = -\frac{1}{2} + \frac{\sqrt{11}i}{2} \)[/tex]. The real zero indicates where the function intersects the x-axis, while the complex zeros affect the curvature and shape of the graph but do not show up as x-axis intersections.
