OF POLYNOMIAL FUNCTIONS Identify the degree, leading coefficient, zeros, end behavior, and create a sketch of the fu without the use of technology. (An accurate \( y \)-intercept is not needed for the sketch). Part 1 \( \begin{array}{ll}\text { (1) } p(r)=-2(r-1)(r+2)(r-2) & \text { (2.) } f(t)=3 t(t-3)(t+4)\end{array} \)

Let's analyze each polynomial function step by step. ### Part 1: Analyzing \( p(r) = -2(r-1)(r+2)(r-2) \) 1. **Degree**: - The degree of a polynomial is determined by the highest power of the variable. Here, we have three linear factors: \( (r-1) \), \( (r+2) \), and \( (r-2) \). Therefore, the degree is \( 3 \). 2. **Leading Coefficient**: - The leading coefficient is the coefficient of the term with the highest degree. In this case, the leading coefficient is \( -2 \) (from the factor \( -2 \) in front of the polynomial). 3. **Zeros**: - The zeros of the polynomial are found by setting each factor equal to zero: - \( r - 1 = 0 \) → \( r = 1 \) - \( r + 2 = 0 \) → \( r = -2 \) - \( r - 2 = 0 \) → \( r = 2 \) - Therefore, the zeros are \( r = 1, -2, 2 \). 4. **End Behavior**: - Since the degree is odd (3) and the leading coefficient is negative, the end behavior is: - As \( r \to -\infty \), \( p(r) \to +\infty \) - As \( r \to +\infty \), \( p(r) \to -\infty \) 5. **Sketch**: - The polynomial will cross the x-axis at the zeros \( r = -2, 1, 2 \). - The graph will rise to the left and fall to the right, reflecting the end behavior. ### Part 2: Analyzing \( f(t) = 3t(t-3)(t+4) \) 1. **Degree**: - Similar to the previous polynomial, we have three linear factors: \( t \), \( (t-3) \), and \( (t+4) \). Therefore, the degree is \( 3 \). 2. **Leading Coefficient**: - The leading coefficient is the coefficient of the term with the highest degree. Here, it is \( 3 \). 3. **Zeros**: - The zeros of the polynomial are found by setting each factor equal to zero: - \( t = 0 \) - \( t - 3 = 0 \) → \( t = 3 \) - \( t + 4 = 0 \) → \( t = -4 \) - Therefore, the zeros are \( t = 0, 3, -4 \). 4. **End Behavior**: - Since the degree is odd (3) and the leading coefficient is positive, the end behavior is: - As \( t \to -\infty \), \( f(t) \to -\infty \) - As \( t \to +\infty \), \( f(t) \to +\infty \) 5. **Sketch**: - The polynomial will cross the x-axis at the zeros \( t = -4, 0, 3 \). - The graph will fall to the left and rise to the right, reflecting the end behavior. ### Summary of Results 1. For \( p(r) = -2(r-1)(r+2)(r-2) \): - Degree: \( 3 \) - Leading Coefficient: \( -2 \) - Zeros: \( r = -2, 1, 2 \) - End Behavior: \( \begin{cases} r \to -\infty & \Rightarrow p(r) \to +\infty \\ r \to +\infty & \Rightarrow p(r) \to -\infty \end{cases} \) 2. For \( f(t) = 3t(t-3)(t+4) \): - Degree: \( 3 \) - Leading Coefficient: \( 3 \) - Zeros: \( t = -4, 0, 3 \) - End Behavior: \( \begin{cases} t \to -\infty & \Rightarrow f(t) \to -\infty \\ t \to +\infty & \Rightarrow f(t) \to +\infty \end{cases} \) ### Sketches - For \( p(r) \), the graph will cross the x-axis at \( -2, 1, 2 \) and will rise to the left and fall to the right. - For \( f(t) \), the graph will cross the x-axis at \( -4, 0, 3 \) and will fall to the left and rise to the right. These sketches can be drawn based on the zeros and end behavior described.