For the polynomial function \( f(x)=x^{4}+4 x^{3}+4 x^{2} \), answer the parts a through \( e \). A. The \( x \)-intercept(s) at which the graph crosses the \( x \)-axis is/are \( \square \) 1. (Type an inleger or a decimal. Uso a comma lo separate answers as needed. Type each answer only once.) B. There are no \( x \)-intercepts at which the graph crosses the \( x \)-axis. At which \( x \)-intercept(s) does the graph touch the \( x \)-axis and furn around? Select the corract chcice bolow end, it nocossary, fit in the answer box to completo your choice. A. The \( x \)-inlerceptis) at which the graph touches the \( x \)-axis and turns around isfare \( -2,0 \). (Type an inloger or a docimal. Use a comma to separate answers as needed Type each answer conly once.) B. There are no \( x \)-infercepts at which the graph toaches the \( x \)-axis and lurns around. c. Find the \( y \)-intercepl. The y intorcept is 0 (Sumplify your answet. Type an inleger or a decimal.) d. Determine whether the graph has y-axis symmetry, origin symutiry, or neither. Chooso the cortect answor below. A. The graph of \( I \) is symmetric about the nrigin. B. The graph of \( I \) is symmetric about the \( y \) axis. C. The graph of \( I \) is neither symmerric aboul the \( y \)-axis nor symmetric aboul the origin.
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Bonus Knowledge
To solve the polynomial function \( f(x) = x^4 + 4x^3 + 4x^2 \), let's analyze it step by step. A. To find the \( x \)-intercepts, we set \( f(x) = 0 \): \[ x^4 + 4x^3 + 4x^2 = 0 \implies x^2(x^2 + 4x + 4) = 0. \] This gives us \( x^2 = 0 \) or the quadratic \( (x+2)^2 = 0 \). Thus, \( x = 0 \) and \( x = -2 \) (with multiplicity 2). The graph crosses the x-axis at \( 0 \). The answer is \( 0 \). B. Since the \( x=-2 \) intercept has an even multiplicity, the graph touches the x-axis but does not cross it. C. The \( y \)-intercept can be found by evaluating \( f(0) \): \[ f(0) = 0^4 + 4(0^3) + 4(0^2) = 0. \] So, the \( y \)-intercept is \( 0 \). D. For symmetry, a function is symmetric about the \( y \)-axis if \( f(-x) = f(x) \). Checking: \[ f(-x) = (-x)^4 + 4(-x)^3 + 4(-x)^2 = x^4 - 4x^3 + 4x^2, \] which is not equal to \( f(x) \). The graph is not symmetric about the y-axis and since there are no odd powers of \( x \) equal in form, it also doesn't show symmetry about the origin. To summarize: - A. The \( x \)-intercept(s) at which the graph crosses the \( x \)-axis is/are \( 0 \). - B. The \( x \)-intercepts at which the graph touches the \( x \)-axis and turns around is \( -2 \). - C. The \( y \)-intercept is \( 0 \). - D. The graph is symmetric about the \( y \)-axis.
