Question

Graph each polynomial function. Factor first if the expression is not in factored form. \(\displaystyle{f{{\left({x}\right)}}}={x}^{{{2}}}{\left({x}-{5}\right)}{\left({x}+{3}\right)}{\left({x}-{1}\right)}\)

Answer 1

\(\displaystyle{f{{\left({x}\right)}}}={x}^{{{2}}}{\left({x}-{5}\right)}{\left({x}+{3}\right)}{\left({x}-{1}\right)}\) The function in factored form The function has four zeros 0,5,-3 and 1. So, the graph of f(x) crossed the x-axis at (0,0), (5,0),(-3,0), and (1,0) To find the y-intercept, substitute 0 for x in f(x) \(\displaystyle{f{{\left({x}\right)}}}={x}^{{{2}}}{\left({x}-{5}\right)}{\left({x}+{3}\right)}{\left({x}-{1}\right)}\)
\(\displaystyle{f{{\left({0}\right)}}}={0}^{{{2}}}{\left({0}-{5}\right)}{\left({0}+{3}\right)}{\left({0}-{1}\right)}\) Substitute 0 for x
\(\displaystyle={0}\) So, the function f(x) crosses the y-axis at (0,0) Step 2 PSKx^{2}\cdot x \cdot x\cdot x=x^{5} The leading coefficient is 1 Since the leading coefficient is positive and the function f(x) of degree 5 (odd degree) So, the end behavior is \(\displaystyle{x}\rightarrow\infty,{f{{\left({x}\right)}}}\rightarrow\infty\)
\(\displaystyle{x}\rightarrow-\infty,{f{{\left({x}\right)}}}\rightarrow-\infty\) See the graph below FZP https://q2a.s3-us-west-1.amazonaws.com/dev/1961020641.jpg"/>