Chapter 1: Fundamentals

Simplify the fractional expression.

$\frac{2x{\left(x+6\right)}^4-x^2\left(4\right){\left(x+6\right)}^3}{{\left(x+6\right)}^8}$

Conjugate Recall that the symbol$\overline{z}$represents the complex conjugate of z. If$z=a+bi\text{and}w=c+di$, show that each statement is true.

$\overline{\overline{z}}=z$

Find the real solutions of the quadratic equation $25x^2+7x+49=0$.

Real and Complex Solutions Find all real and complex solutions of the equation.

$x^3-2x^2+4x-8=0$

A jeweler has three small solid spheres made of gold, of radius 2 mm, 3 mm, and 4 mm. He decides to melt these down and make just one sphere out of them. What will the radius of this larger sphere be?

Use a Special Factoring Formula to factor the expression

$a^3-b^6$

Solve the absolute value inequality. Express the answer using interval notation and graph the solution set.

$\left|x+1\right|\ge 1$

Find the number of real solutions of the equation $x^2-6x+1=0$ by using discriminant.

An equation is given.

(a)Use a graphing calculator to graph the equation in the given viewing rectangle. (b) Find the x- and y-intercepts from the graph.

(c) Verify your answers to part (b) algebraically (from the equation).

$y=-\frac{2}{x^2+1};\left[-5,5\right]\text{by}\left[-3,1\right]$

Recall that the symbol $\overline{z}$represents the complex conjugate of z. If $z=a+bi\text{and}w=c+di$, show that each statement is true.

$z+\overline{z}\text{isarealnumber.}$