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Use Theorem complexfactorization to create a polynomial function with real number coefficients which has all of the desired characteristics. You
may leave the polynomial in factored form.
The zeros of \(f\) are \(c = \pm 2\) and \(c = \pm 1\).
The leading term of \(f(x)\) is \(117x^4\).
\(f(x) = \answer {117(x+2)(x-2)(x+1)(x-1)}\)
Use Theorem complexfactorization to create a polynomial function with real number coefficients which has all of the desired characteristics. You
may leave the polynomial in factored form.
The zeros of \(p\) are \(c=1\) and \(c = 3\).
\(c=3\) is a zero of multiplicity 2.
The leading term of \(p(z)\) is \(-5z^3\).
\(p(z)= \answer {-5(z-1)(z-3)^2}\)
Use Theorem complexfactorization to create a polynomial function with real number coefficients which has all of the desired characteristics. You
may leave the polynomial in factored form.
The solutions to \(g(t) = 0\) are \(t = \pm 3\) and \(t=6\).
The leading term of \(g(t)\) is \(7t^4\).
The point \((-3,0)\) is a local minimum on the graph of \(y=g(t)\).
\(g(t) = \answer {7(t+3)^2(t-3)(t-6)}\)
Use Theorem complexfactorization to create a polynomial function with real number coefficients which has all of the desired characteristics. You
may leave the polynomial in factored form.
The solutions to \(f(x) =0\) are \(x = \pm 3\), \(x=-2\), and \(x=4\).
The leading term of \(f(x)\) is \(-x^5\).
The point \((-2, 0)\) is a local maximum on the graph of \(y=f(x)\).
Use Theorem complexfactorization to create a polynomial function with real number coefficients which has all of the desired characteristics. You
may leave the polynomial in factored form.
\(p\) has exactly three \(z\)-intercepts: \((-6,0)\), \((1,0)\) and \((117,0)\).
The graph of \(y=p(z)\) crosses through the \(z\)-axis at \((1,0)\).
\(p(z) = \answer {(z+6)^2(z-1)(z-117)}\) where \(a\) can be any real number as long as \(a<0\)
Use Theorem complexfactorization to create a polynomial function with real number coefficients which has all of the desired characteristics. You
may leave the polynomial in factored form.
The zeros of \(g\) are \(c=\pm 1\) and \(c = \pm i\).
The leading term of \(g(t)\) is \(42t^4\).
\(g(t) = \answer {42(t-1)(t+1)(t-i)(t+i)}\)
Use Theorem complexfactorization to create a polynomial function with real number coefficients which has all of the desired characteristics. You
may leave the polynomial in factored form.
\(c=2i\) is a zero.
the point \((-1,0)\) is a local minimum on the graph of \(y=f(x)\).
the leading term of \(f(x)\) is \(117x^4\).
\(f(x) = \answer {117(x+1)^2(x-2i)(x+2i)}\)
Use Theorem complexfactorization to create a polynomial function with real number coefficients which has all of the desired characteristics. You
may leave the polynomial in factored form.
The solutions to \(p(z) = 0\) are \(z = \pm 2\) and \(z=\pm 7i\).
The leading term of \(p(z)\) is \(-3z^5\).
The point \((2,0)\) is a local maximum on the graph of \(y=p(z)\).
\(p(z) = -3(z-2)^2(z+2)(z-7i)(z+7i)\)
Use Theorem complexfactorization to create a polynomial function with real number coefficients which has all of the desired characteristics. You
may leave the polynomial in factored form.
\(g\) is degree \(5\).
\(t=6\), \(t = i\) and \(t = 1-3i\) are zeros of \(g\).
\(g(t) = a(t-6)(t-i)(t+i)(t-(1-3i))(t-(1+3i))\) where \(a\) is any real number, \(a < 0\)
Use Theorem complexfactorization to create a polynomial function with real number coefficients which has all of the desired characteristics. You
may leave the polynomial in factored form.
The leading term of \(f(x)\) is \(-2x^3\).
\(c=2i\) is a zero.
\(f(0) = -16\).
\(f(x) = -2(x-2i)(x+2i)(x+2)\)
Find a possible formula for the polynomial function given its graph. You may leave the polynomial in factored
form.
\(f(x) = x(x+6)(x-6)\)
Find a possible formula for the polynomial function given its graph. You may leave the polynomial in factored
form.
\(g(t) = t(t+2)^3\)
Find a possible formula for the polynomial function given its graph. You may leave the polynomial in factored
form.
\(y = p(z)\)
\(p(z) = -2(z+1)(z-2)^2\)
Find a possible formula for the polynomial function given its graph. You may leave the polynomial in factored
form.
Find a possible formula for the polynomial function given its graph. You may leave the polynomial in factored
form.
\(F(s) =-s(s+2)^2\)
Find a possible formula for the polynomial function given its graph. You may leave the polynomial in factored
form.
\(G(t) = t^3(t+2)^2\)
With help from your classmates, choose several nonzero complex numbers \(z\), find their complex conjugates \(\overline {z}\). Plot each pair \(z\)
and \(\overline {z}\) in the Complex Plane. What appears to be the relationship between these numbers geometrically? State and prove a
general result.
If \(z = a+bi\), then \(z\) corresponds to the point \((a,b)\) in the \(xy\)-plane. Hence, \(\overline {z} = \overline {a+bi} = a-bi\) corresponds to the point \((a,-b)\). Hence, the points corresponding to \(z\) and \(\overline {z}\)
are reflections about the \(x\)-axis.
With help from your classmates, choose several nonzero complex numbers \(z\) and find \(-z\). Plot each pair \(z\) and \(-z\) in the Complex
Plane. What appears to be the relationship between these numbers geometrically? State and prove a general
result.
If \(z = a+bi\), then \(z\) corresponds to the point \((a,b)\) in the \(xy\)-plane. Hence, \(-z =-(a+bi) = -a-bi\) corresponds to the point \((-a,-b)\). Hence, the points corresponding to \(z\) and \(-z\)
are reflections through the origin.
With help from your classmates, choose several different complex numbers \(z\) and find the product of \(i\) and \(z\), \(iz\). Plot each pair of \(z\)
and \(iz\) in the Complex Plane. In each case, show the line containing the origin and the point corresponding
to \(z\) is perpendicular (See Theorem parallelperpendicularslopetheorem in Section AppLines of you need a refresher on how to do this.) to the line
containing the origin and the point corresponding to \(iz\). Show this result holds in general for every nonzero complex
number.
If \(z = a+bi\), then \(z\) corresponds to the point \((a,b)\) in the \(xy\)-plane. Writing out the product \(iz\), we get: \(iz =i(a+bi) = ia+bi^2 = ia - b = -b+ia\). Hence, \(iz\) corresponds to the point \((-b,a)\). If \(z \neq 0\), then
neither \(a\) nor \(b\) is \(0\) (do you see why?) Hence, the slope of the line containing \((0,0)\) and \((a,b)\) is \(\frac {b}{a}\) and the slope of the line containing \((0,0)\) and \((-b,a)\) is \(-\frac {a}{b}\).
Per Theorem parallelperpendicularslopetheorem, since the slopes of these lines are negative reciprocals, the lines themselves are perpendicular. (We’ll be
able to show in Section PolarComplex that, more precisely, multiplication by \(i\) rotates the complex number counter-clockwise by \(90^{\circ }\).)
Given a complex number \(z = a+bi\), we define the modulus of \(z\), \(|z|\), by \(|z| = \sqrt {a^2+b^2}\). With help from your classmates, calculate \(|z|\) for
several different complex numbers, \(z\). What does \(z\) measure geometrically? Show that if \(x\) is a real number, then the
modulus of \(x\) is the same as the absolute value of \(x\), and comment how all this relates to Definition absvaldistdefn in Section
AppAbsValEqIneq.
\(|z| = \sqrt {a^2+b^2}\) measures the distance from the origin to the point \((a,b)\). Hence, \(|z|\) measures the distance from \(z\) to \(0\) in the Complex Plane. This is
exactly how \(|x|\) is defined in Definition absvaldistdefn in Section AppAbsValEqIneq. In that section, however, the only part of the Complex Plane under discussion
is the real number line.
Let \(z\) and \(w\) be arbitrary complex numbers. Show that \(\overline {z} \, \overline {w} = \overline {zw}\) and \(\overline {\overline {z}} = z\).
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