1) Solve the polynomial over the set of Complex Numbers: \(0={{x}^{3}}+2{{x}^{2}}-11x-12\).
2) Solve the polynomial over the set of Complex Numbers: \(0={{x}^{3}}-{{x}^{2}}+4x-4\).
3) Find all roots of \(P(x)=x^3+2x^2-8x-16\).
4) Use synthetic substitution to find \(a\) and \(b\) if both \(1\) and \(-6\) are both zeros of \(y=x^3+2x^2-ax+b\).
For Problems #5-6, use the function \(f(x)=x^4-2x^3+21x^2-32x+80\).
5) Find \(f(4i)+2(f(i))-f(1+2i)\).
6) Find all roots of the polynomial. (Hint: your answer to #5 should make this problem a lot easier.)
7) The fundamental theorem of algebra guarantees that \( f(x) = x^3 -1 \) has three complex roots. Clearly the only real root is \( x= 1 \). Find the other roots.
8) \( g(x) = x^5 + a x^4 + 50 x^3 + b x^2 + cx + d \) has one real positive root with multiplicity 5. Find the value of \( a + b + c + d \).
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2) Solve the polynomial over the set of Complex Numbers: \(0={{x}^{3}}-{{x}^{2}}+4x-4\).
3) Find all roots of \(P(x)=x^3+2x^2-8x-16\).
4) Use synthetic substitution to find \(a\) and \(b\) if both \(1\) and \(-6\) are both zeros of \(y=x^3+2x^2-ax+b\).
For Problems #5-6, use the function \(f(x)=x^4-2x^3+21x^2-32x+80\).
5) Find \(f(4i)+2(f(i))-f(1+2i)\).
6) Find all roots of the polynomial. (Hint: your answer to #5 should make this problem a lot easier.)
7) The fundamental theorem of algebra guarantees that \( f(x) = x^3 -1 \) has three complex roots. Clearly the only real root is \( x= 1 \). Find the other roots.
8) \( g(x) = x^5 + a x^4 + 50 x^3 + b x^2 + cx + d \) has one real positive root with multiplicity 5. Find the value of \( a + b + c + d \).
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