For the pulynomial tunction \( f(x)=x^{4}+4 x^{3}+4 x^{2} \), answor the parts a ttrougn \( e \). b. Find the \( x \)-intorcept(s) State whether the graph crosses the \( x \)-axcs, or touches the \( x \)-axis and turns around, al each intercepl The \( x \)-inlercept(s) is/are D. - 2 . (Type an integer or a decimal. Use a currma to separale answers as needed Type each answor only oncc.) At which \( x \)-intercept(s) does the grapli cruss the \( x \)-axis? Salect tho coifect choicn below and, it nocossary, fill in the answer box to complele your choice. A. The \( x \)-intercept(s) at which the graph crosses the \( x \)-axis is/are \( \square \) (Type an integet or a decimal thon a comma to soparato answers as needed. Type each answet only onco.) B. Thare are no \( x \)-intercepls al which tho greph crossos the \( x \)-axis. At which \( x \)-intercepl(s) does the graph touch the \( x \)-axis and lum around? Select the confect choice below end, it necessary, fill in the answer box to complete your choice A. The \( x \)-irtercepl(s) at which the graph louches the \( x \)-axis andfurns around is/are \( -2,0 \) (Type an inleger or a decimal. Use a comma to soparalo answors as noeded. Type each answer only unce.) B. There are no \( x \)-intorcepls al which the graph touches tho \( x \)-axis and turns arcund. c. Find the \( y \)-intercept. The \( y \)-intercept is \( \square \) . (Simplify your answer. Type an inleger or a decimal.)

To solve the problem step by step, we will analyze the polynomial function \( f(x) = x^4 + 4x^3 + 4x^2 \). ### Part a: Finding the \( x \)-intercepts 1. **Finding the \( x \)-intercepts**: The \( x \)-intercepts occur where \( f(x) = 0 \). We need to solve the equation: \[ x^4 + 4x^3 + 4x^2 = 0 \] 2. **Factoring the polynomial**: We can factor out the common term \( x^2 \): \[ x^2(x^2 + 4x + 4) = 0 \] This gives us: \[ x^2 = 0 \quad \text{or} \quad x^2 + 4x + 4 = 0 \] 3. **Solving \( x^2 = 0 \)**: \[ x = 0 \] 4. **Solving \( x^2 + 4x + 4 = 0 \)**: This can be factored as: \[ (x + 2)^2 = 0 \] Thus, we have: \[ x + 2 = 0 \implies x = -2 \] 5. **Summary of \( x \)-intercepts**: The \( x \)-intercepts are \( x = 0 \) and \( x = -2 \). ### Part b: Analyzing the behavior at the \( x \)-intercepts 1. **At \( x = 0 \)**: Since the factor \( x^2 \) is squared, the graph touches the \( x \)-axis and turns around. 2. **At \( x = -2 \)**: Since the factor \( (x + 2)^2 \) is also squared, the graph touches the \( x \)-axis and turns around. 3. **Conclusion**: - The \( x \)-intercepts are \( -2, 0 \). - The graph touches the \( x \)-axis and turns around at both \( -2 \) and \( 0 \). ### Part c: Finding the \( y \)-intercept 1. **Finding the \( y \)-intercept**: The \( y \)-intercept occurs where \( x = 0 \): \[ f(0) = 0^4 + 4(0)^3 + 4(0)^2 = 0 \] ### Final Answers - The \( x \)-intercept(s) is/are: **-2, 0** - The \( x \)-intercept(s) at which the graph crosses the \( x \)-axis is/are: **There are no \( x \)-intercepts at which the graph crosses the \( x \)-axis.** - The \( x \)-intercept(s) at which the graph touches the \( x \)-axis and turns around is/are: **-2, 0** - The \( y \)-intercept is: **0**