Exercises

1.
Given that when x2 - 3x + 5k is divided by x + 4 the remainder is 9, find the value of k using Divide on the Polynomials submenu and Solve + Exact.BITMAPSETAnswer0.2171in0.2006in0ina1

2.
Define functions f (x) = x3 + x ln x and g(x) = x + ex. Evaluate f (g(x)), g(f (x)), f (x)g(x), and f (x) + g(x). BITMAPSETAnswer0.2171in0.2006in0ina2

3.
Find the equation of the line passing through the two points $\left(\vphantom{ x_{1},y_{1}}\right.$x1, y1$\left.\vphantom{ x_{1},y_{1}}\right)$, $\left(\vphantom{ x_{2},y_{2}}\right.$x2, y2$\left.\vphantom{ x_{2},y_{2}}\right)$. BITMAPSETAnswer0.2171in0.2006in0ina3

4.
Find the equation of the line passing through the two points $\left(\vphantom{ 2,5}\right.$2, 5$\left.\vphantom{ 2,5}\right)$, $\left(\vphantom{ 3,-7}\right.$3, - 7$\left.\vphantom{ 3,-7}\right)$.BITMAPSETAnswer0.2171in0.2006in0ina4

5.
Find the equation of the line passing through the two points $\left(\vphantom{ 1,2}\right.$1, 2$\left.\vphantom{ 1,2}\right)$, $\left(\vphantom{ 2,4}\right.$2, 4$\left.\vphantom{ 2,4}\right)$.BITMAPSETAnswer0.2171in0.2006in0ina5

6.
Find the slope of the line determined by the equation sx + ty = c.BITMAPSETAnswer0.2171in0.2006in0ina6

7.
Find the center and semi-axes of the ellipse  x2 +4y2 + 96x - 8y + 84 = 0.BITMAPSETAnswer0.2171in0.2006in0ina7

8.
Factor the difference of powers xn - yn for several values of n, and deduce a general formula.BITMAPSETAnswer0.2171in0.2006in0ina8

9.
Applying Factor to x2 + $\left(\vphantom{ \sqrt{5}%
-3}\right.$$\sqrt{{5}%
}$ - 3$\left.\vphantom{ \sqrt{5}%
-3}\right)$x - 3$\sqrt{{5}}$ gives the factorization

x2 + $\displaystyle \left(\vphantom{ \sqrt{5}-3}\right.$$\displaystyle \sqrt{{5}}$ - 3$\displaystyle \left.\vphantom{ \sqrt{5}-3}\right)$x - 3$\displaystyle \sqrt{{5}}$ =  $\displaystyle \left(\vphantom{ x+\sqrt{5}}\right.$x + $\displaystyle \sqrt{{5}}$$\displaystyle \left.\vphantom{ x+\sqrt{5}}\right)$$\displaystyle \left(\vphantom{
x-3}\right.$x - 3$\displaystyle \left.\vphantom{
x-3}\right)$

showing that the system can factor some polynomials with irrational roots. However, applying Factor to  x2 - 3 and x3 +3x2 - 5x + 1 does not do anything. Find a way to get Scientific Notebook to factor these polynomials.BITMAPSETAnswer0.2171in0.2006in0ina9

10.
If principal P is invested at an interest rate r compounded annually, in t years it grows to an amount A given by A = P$\left(\vphantom{ 1+r}\right.$1 + r$\left.\vphantom{ 1+r}\right)^{{t}}_{}$. Find the number of years required for the amount to double at interest rates of 3%, 5%, 8%, and 10%.

BITMAPSETAnswer0.2171in0.2006in0ina10

11.
Find all real and complex solutions to the system of equations
2x2 - y = 1  
x + 3y3 = 4  

BITMAPSETAnswer0.2171in0.2006in0ina11

12.
The curves determined by quadratic equations of the form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 are often referred to as conic sections, as each can be described as the intersection of a plane with a cone. The discriminant of the expression Ax2 + Bxy + Cy2 + Dx + Ey + F is the number

$\displaystyle \bigtriangleup$  = B2 - 4AC

The type of curve determined by the quadratic equation depends on the sign of the discriminant. In nondegenerate cases, the quadratic equation gives

$\displaystyle \fbox{$
\begin{array}{r}
\text{an ellipse if }\bigtriangleup \;...
...up \;=\;0 \\
\text{a hyperbola if }\bigtriangleup \;>\;0
\end{array}
$}
$

where a circle is considered as a special case of an ellipse. The degenerate cases give a pair of lines, a line, or a point, or they have imaginary solutions.

A rotation of one or both axes in the plane leads to a change of variables such as the change

u = ax + by  
v = cx + dy  

from $\left(\vphantom{ x,y}\right.$x, y$\left.\vphantom{ x,y}\right)$ to $\left(\vphantom{ u,v}\right.$u, v$\left.\vphantom{ u,v}\right)$. Show that the sign of the discriminant (and thus the basic shape of the curve) is invariant under such changes for a quadratic equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 if adbc.BITMAPSETAnswer0.2171in0.2006in0ina12