For Questions 1-2 sketch a graph of each of the following polynomials. Be sure to list out the “important stuff” and to account for multiplicities, leading coefficients, and y-intercepts.
1) \(f(x)=(x+6)({{x}^{2}}+4){{(3-x)}^{2}}(1-x)\)
2) \( f(x)=4{{(x-2)}^{3}}{{(3+x)}^{2}}{{(1-x)}^{3}}\)
3) Given the following graph, write the equation for the polynomial.
1) \(f(x)=(x+6)({{x}^{2}}+4){{(3-x)}^{2}}(1-x)\)
2) \( f(x)=4{{(x-2)}^{3}}{{(3+x)}^{2}}{{(1-x)}^{3}}\)
3) Given the following graph, write the equation for the polynomial.
4) Given the following graph, write the equation for the polynomial.
5) Write the equation for the polynomial in factored form. \(P(x)={{x}^{4}}+4{{x}^{3}}-7{{x}^{2}}-22x+24\).
6) The only zeros of a polynomial function \(P(x)\) are \(x = 9\), \(x = -6\), \(x = 0\), and \(x = 2\):
a) What are the factors of \(P(x)\)?
b) How many \(x\)-intercepts are there for \(P(x)\)?
c) Assume that no root has multiplicity and the leading coefficient is positive. Sketch the graph of this polynomial and
write it in expanded form.
7) \(R(x) \) is a degree 4 polynomial whose graph crosses the x-axis exactly twice at rational values. If \( p(x) \) and \( q(x) \) are quadratic polynomials such that \( R(x) = p(x) q(x) \), what can you conclude about the discriminants of \( p(x) \) and \( q(x) \).
8) \( f(x) \) is a polynomial with root \(x=a\) and no constant term, and \( g(x) \) is a polynomial with root \(x=b\). Which of the following polynomials are guaranteed to have \(a\) as a root, and which are guaranteed to have \(b\) as a root?
\( f(x) g(x) \)
\( f(x)^2 \)
\( f(x) + g(x) \)
\( f( g(x) ) \)
\( g(x^2) \)
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