Question
Which graph is a possible sketch of the polynomial f(x)=x3+5x2+6x
?(1 point)
Responses
?(1 point)
Responses
Answers
GPT-4o mini
To sketch the polynomial function \( f(x) = x^3 + 5x^2 + 6x \), we can first analyze its properties:
1. **Degree and Leading Coefficient**: The function is a cubic polynomial (degree 3) with a positive leading coefficient (1). This means as \( x \) approaches positive infinity, \( f(x) \) will also approach positive infinity, and as \( x \) approaches negative infinity, \( f(x) \) will approach negative infinity.
2. **Roots/Intercepts**: We can factor \( f(x) \):
\[
f(x) = x(x^2 + 5x + 6) = x(x+2)(x+3)
\]
This tells us that the roots of the polynomial are \( x = 0, -2, -3 \). Thus, the graph will intersect the x-axis at these points.
3. **Behavior at the Roots**: At each of these roots:
- At \( x = 0 \), the graph touches the axis (cross point).
- At \( x = -2 \) and \( x = -3 \), the graph crosses the x-axis.
4. **Y-Intercept**: When \( x = 0 \):
\[
f(0) = 0^3 + 5 \cdot 0^2 + 6 \cdot 0 = 0
\]
So, the y-intercept is also at \( (0, 0) \).
5. **End Behavior**: The graph will start in the third quadrant when \( x \to -\infty \), cross the x-axis at the roots mentioned, and continue to rise in the first quadrant as \( x \to +\infty \).
Based on this analysis, a possible sketch of the graph should show:
- Three roots at \( x = 0, -2, -3 \)
- The graph should be in the third quadrant for negative \( x \) and move into the first quadrant for positive \( x \).
If you have options to choose from, look for a graph that reflects these characteristics.
1. **Degree and Leading Coefficient**: The function is a cubic polynomial (degree 3) with a positive leading coefficient (1). This means as \( x \) approaches positive infinity, \( f(x) \) will also approach positive infinity, and as \( x \) approaches negative infinity, \( f(x) \) will approach negative infinity.
2. **Roots/Intercepts**: We can factor \( f(x) \):
\[
f(x) = x(x^2 + 5x + 6) = x(x+2)(x+3)
\]
This tells us that the roots of the polynomial are \( x = 0, -2, -3 \). Thus, the graph will intersect the x-axis at these points.
3. **Behavior at the Roots**: At each of these roots:
- At \( x = 0 \), the graph touches the axis (cross point).
- At \( x = -2 \) and \( x = -3 \), the graph crosses the x-axis.
4. **Y-Intercept**: When \( x = 0 \):
\[
f(0) = 0^3 + 5 \cdot 0^2 + 6 \cdot 0 = 0
\]
So, the y-intercept is also at \( (0, 0) \).
5. **End Behavior**: The graph will start in the third quadrant when \( x \to -\infty \), cross the x-axis at the roots mentioned, and continue to rise in the first quadrant as \( x \to +\infty \).
Based on this analysis, a possible sketch of the graph should show:
- Three roots at \( x = 0, -2, -3 \)
- The graph should be in the third quadrant for negative \( x \) and move into the first quadrant for positive \( x \).
If you have options to choose from, look for a graph that reflects these characteristics.