Chapter 1: Fundamentals

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.

$\overline{zw}=\overline{z}\cdot \overline{w}$

The equations of two lines are given. Determine whether the lines are parallel, perpendicular, or neither. (Lesson 1.10)

$6y-2x=5;$$2y+6x=1$

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).

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

Express the quantity without using absolute value

$\left|\pi -3\right|$

Simplify the fractional expression.

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

A large plywood box has a volume of. Its length is 9 ft greater than its height, and its width is 4 ft less than its height. What are the dimensions of the box.

Using Slopes Verify the given geometric property.

Use slopes to show that$A\left(1,1\right),B\left(7,4\right),C\left(5,10\right)$ and $D\left(-1,7\right)$are vertices of a parallelogram.

Find the real and complex solution of the equation $x^4-256=0$.

Find the real solutions of the quadratic equation $10y^2-16y+5=0$.

Use a Special Factoring Formula to factor the expression

$27x^3+y^3$