Question
f\left(x\right)=x\left(x-1\right)\left(2x+4\right)^2
Answers
GPT-4o mini
To analyze the function \( f(x) = x(x-1)(2x+4)^2 \), let's go through a few steps:
1. **Identify the roots (zeros) of the function**:
- The factors of \( f(x) \) are \( x \), \( x-1 \), and \( (2x+4)^2 \).
- Setting each factor equal to zero:
- \( x = 0 \)
- \( x - 1 = 0 \) → \( x = 1 \)
- \( (2x + 4)^2 = 0 \) → \( 2x + 4 = 0 \) → \( 2x = -4 \) → \( x = -2 \)
The roots of the function are \( x = 0, 1, -2 \).
2. **Determine the multiplicity of the roots**:
- The root at \( x = 0 \) has multiplicity 1.
- The root at \( x = 1 \) has multiplicity 1.
- The root at \( x = -2 \) has multiplicity 2 (because it appears squared in the factor).
3. **Behavior at each root**:
- Since \( x = 0 \) and \( x = 1 \) have odd multiplicities (1), the graph will cross the x-axis at these points.
- Since \( x = -2 \) has even multiplicity (2), the graph will touch the x-axis but not cross it at this point.
4. **End behavior**:
As \( x \to -\infty \), the leading term is negative due to \( 2x \) being dominant in the square term and also \( x(x-1) \to -\infty \) (as \( x \to -\infty \)), hence \( f(x) \to -\infty \).
As \( x \to +\infty \), all terms become positive, so \( f(x) \to +\infty \).
5. **Summary of the function characteristics**:
- Roots: \( x = -2 \) (multiplicity 2), \( x = 0 \) (multiplicity 1), \( x = 1 \) (multiplicity 1).
- End behavior: \( f(x) \to -\infty \) as \( x \to -\infty \) and \( f(x) \to +\infty \) as \( x \to +\infty \).
- The function will cross the x-axis at \( x = 0 \) and \( x = 1 \) while touching but not crossing at \( x = -2 \).
If you need further exploration like derivatives, critical points, or graphing the function, please let me know!
1. **Identify the roots (zeros) of the function**:
- The factors of \( f(x) \) are \( x \), \( x-1 \), and \( (2x+4)^2 \).
- Setting each factor equal to zero:
- \( x = 0 \)
- \( x - 1 = 0 \) → \( x = 1 \)
- \( (2x + 4)^2 = 0 \) → \( 2x + 4 = 0 \) → \( 2x = -4 \) → \( x = -2 \)
The roots of the function are \( x = 0, 1, -2 \).
2. **Determine the multiplicity of the roots**:
- The root at \( x = 0 \) has multiplicity 1.
- The root at \( x = 1 \) has multiplicity 1.
- The root at \( x = -2 \) has multiplicity 2 (because it appears squared in the factor).
3. **Behavior at each root**:
- Since \( x = 0 \) and \( x = 1 \) have odd multiplicities (1), the graph will cross the x-axis at these points.
- Since \( x = -2 \) has even multiplicity (2), the graph will touch the x-axis but not cross it at this point.
4. **End behavior**:
As \( x \to -\infty \), the leading term is negative due to \( 2x \) being dominant in the square term and also \( x(x-1) \to -\infty \) (as \( x \to -\infty \)), hence \( f(x) \to -\infty \).
As \( x \to +\infty \), all terms become positive, so \( f(x) \to +\infty \).
5. **Summary of the function characteristics**:
- Roots: \( x = -2 \) (multiplicity 2), \( x = 0 \) (multiplicity 1), \( x = 1 \) (multiplicity 1).
- End behavior: \( f(x) \to -\infty \) as \( x \to -\infty \) and \( f(x) \to +\infty \) as \( x \to +\infty \).
- The function will cross the x-axis at \( x = 0 \) and \( x = 1 \) while touching but not crossing at \( x = -2 \).
If you need further exploration like derivatives, critical points, or graphing the function, please let me know!