1. Perform the requested divisions. Find the quotient and remainder and verify the Remainder Theorem by computing \( p(a) \).
a) Divide \( p(x)=x^{2}-5 x+8 \) by \( x+4 \)
b) Divide \( p(x)=2 x^{3}-7 x^{2}+x+4 \) by \( x-4 \)
c) Divide \( p(x)=1-x^{4} \) by \( x-1 \)
d) Divide \( p(x)=x^{5}-2 x^{2}-3 \) by \( x+1 \)
2. Given that \( p(4)=0 \), factor \( p(x)=2 x^{3}-11 x^{2}+10 x+8 \) as completely as possible.
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3. Given that \( r(x)=4 x^{3}-x^{2}-36 x+9 \) and \( r\left(\frac{1}{4}\right)=0 \), find the remaining zeros of \( r(x) \).
4. Given that 3 is a double zero of \( p(x)=x^{4}-3 x^{3}-19 x^{2}+87 x-90 \), find all the zeros of \( p(x) \).
5. a) Write the general polynomial \( p(x) \) whose only zeros are 1,2 and 3 , with multiplicity 3,2 and 1 respectively. What is its degree?
b) Find \( p(x) \) described in part (a) if \( p(0)=6 \).
6. If \( 2-3 i \) is a root of \( p(x)=2 x^{3}-5 x^{2}+14 x+39 \), find the remaining zeros of \( \mathrm{p}(x) \).
7. Determine the rational zeros of the polynomials
a) \( p(x)=x^{3}-4 x^{2}-7 x+10 \)
b) \( p(x)=2 x^{3}-5 x^{2}-28 x+15 \)
c) \( p(x)=6 x^{3}+x^{2}-4 x+1 \)
8. Find the domain and the real zeros of the given function.
a) \( f(x)=\frac{3}{x^{2}-25} \)
b) \( g(x)=\frac{x-3}{x^{2} 4 x-12} \)
c) \( f(x)=\frac{(x-3)^{2}}{x^{3}-3 x^{2}+2 x} \)
d) \( f(x)=\frac{x^{2}-16}{x^{2}+4} \)
9. Sketch the graph of
a) \( f(x)=\frac{1-x}{x-3} \)
b) \( f(x)=\frac{x^{2}+1}{x} \)
c) \( f(x)=\frac{1}{x}+2 \)
d) \( f(x)=\frac{x^{2}}{x^{2}-4} \)
10. Determine the behavior of \( f(x)=\frac{x^{3}-8 x-3}{x-3} \) when \( x \) is near 3 .
11. The graph of any rational function in which the degree of the numerator is exactly one more than the degree of the denominator will have an oblique (or slant) asymptote.
a) Use long division to show tha 86 / 217
\[
y=f(x)=\frac{x^{2}-x+6}{x-2}=x+1+\frac{8}{x-2}
\]
b) Show th for the graph and sketch tr
