State what the fundamental theorem of algebra guarantees for a non-constant complex polynomial.

How many complex roots counted with multiplicity does a degree \(6\) polynomial have?

If \(p(z)=(z-2)^3(z+i)\), what is the degree of \(p\)?

In \(p(z)=(z-1)^2(z+3)\), what is the multiplicity of the root \(z=1\)?

List the roots of \(p(z)=(z-1)(z+2)(z-i)\).

Factor \(z^2+1\) over \(\mathbb C\).

A real-coefficient polynomial has root \(3-2i\). What other root must it have?

How many roots remain after factoring a linear factor from a degree \(5\) polynomial?

Use the theorem to predict the number of roots of \(z^4+z+1=0\) over \(\mathbb C\).

Factor \(z^3-1\) completely over \(\mathbb C\).

Explain why \(z^4+4\) must factor into four linear factors over \(\mathbb C\).

If \(p(z)\) is a real cubic and \(1+i\) is a root, explain why at least one real root exists.

Find the missing root of a monic cubic with roots \(2+i\), \(2-i\), and one more root, if the constant term is \(-10\).

Solve \(z^4-5z^2+4=0\) and count the roots with multiplicity.

Can a real polynomial of degree \(5\) have exactly two non-real roots counted with multiplicity? Explain.

Can a real polynomial of degree \(4\) have exactly one non-real root? Explain.

Find a real monic quartic whose roots are \(i\), \(-i\), \(2\), and \(-3\).

A learner says the fundamental theorem of algebra gives a formula for the roots of every polynomial. Correct the statement.

Explain why the fundamental theorem of algebra does not apply to \(e^z-1=0\).

Prove that a monic degree \(n\) polynomial with roots \(z_1,\ldots,z_n\) counted with multiplicity factors as \(\prod_{j=1}^n(z-z_j)\).

If \(F(x)=\int_2^x (t^2+1)\,dt\), find \(F'(x)\).

State the evaluation theorem for \(\int_a^b f(x)\,dx\) using an antiderivative \(F\).

If \(G(x)=\int_0^x \cos t\,dt\), find \(G'(x)\).

If \(H(x)=\int_x^5 t^3\,dt\), find \(H'(x)\).

Use the Fundamental Theorem of Calculus to evaluate \(\int_0^2 3x^2\,dx\).

Evaluate \(\int_1^4 (2x-3)\,dx\) using an antiderivative.

Find \(\frac{d}{dx}\int_1^x (4t-7)\,dt\).

Find \(\frac{d}{dx}\int_0^{x^2} (1+t)\,dt\).

Evaluate \(\int_0^1 (6x^2-4x+1)\,dx\).

Let \(A(x)=\int_3^x \sqrt{1+t^2}\,dt\). Find \(A'(2)\).

Let \(S(t)=\int_0^t v(s)\,ds\), where \(v(s)=3s^2+2\) is velocity in metres per second. Find \(S'(t)\) and interpret it.

Show that \(F(x)=\int_0^x 2t\,dt\) equals \(x^2\), then verify \(F'(x)=2x\).

Find \(\frac{d}{dx}\int_{x}^{x^2} t^2\,dt\).

Find \(\frac{d}{dx}\int_1^{3x-2} \sin t\,dt\).

Find \(a\) if \(\int_0^a 2x\,dx=18\) and \(a>0\).

For what \(x\) is \(\frac{d}{dx}\int_0^x (t^2-4)\,dt=0\)?

Let \(F(x)=\int_0^x (t-k)\,dt\). Find \(k\) if \(F'(5)=3\).

A student claims \(\frac{d}{dx}\int_0^{x^2} e^t\,dt=e^{x^2}\). Explain the missing step and give the correct derivative.

Define \(Q(x)=\int_2^x f(t)\,dt\), where \(f\) is continuous and \(f(x)>0\) for all \(x\). Explain why \(Q\) is increasing.

Let \(R(x)=\int_x^{2x} f(t)\,dt\), where \(f\) is continuous. Derive a formula for \(R'(x)\).