Homework 3 Directions:
Show each step of your work and fully simplify each expression.
Turn in your answers in class on a physical piece of paper.
Staple multiple sheets together. Feel free to use Desmos for graphing.
Answer the following:
Factor the following expressions.
$2x^2yz + 7xyz + 3yz$
$4a^2 - 9b^2$
$x^4 - 1 \ $ (hint: Let $y = x^2$)
$(x^2 + 1)^2 - 7(x^2 + 1) + 10$
$x^3 + 4x^2 + x + 4$
$(x+2)(x-1)^2 - 3(x+2)(x-1) + 2(x+2)$
Perform the indicated operation and/or fully simplify. Get rid of all negative exponents.
$\dfrac{4(x+3)(x-1)}{2(x-1)}$
$\dfrac{x^2 + 2x + 1}{x^2 - 1}$
$\dfrac{x^2h + 2xh + h}{h}$
$2 + \dfrac{1}{x + 3}$
$\dfrac{(x+h)^2 - x^2}{h}$
$\dfrac{x^2 + 7x + 12}{x^2 + 3x + 2} \cdot \dfrac{x^2 + 5x + 6}{x^2 + 6x + 9}$
$\dfrac{1}{x + 1} - \dfrac{1}{x - 1}$
$\dfrac{1 + \dfrac{1}{x}}{\dfrac{1}{x} - 2}$
$\dfrac{x}{x + 3} - \dfrac{18}{x^2 - 9}$
$\dfrac{1}{x-1} + \dfrac{1}{x} - \dfrac{1}{x-1}$
$\dfrac{x^{-1} + y^{-1}}{4} \ \ $ (hint: use definition of negative exponent. This is a compound fraction)
$\dfrac{\dfrac{1}{\sqrt{x+h}} - \dfrac{1}{\sqrt{x}}}{h} \ \ $ (hint: get rid of the compound fraction, rationalize the numerator, and cancel common factors)
$\dfrac{\frac{1}{2x + 1}}{\frac{1}{2x^2 + 5x + 2}}$
Rationalize the denominator in $\dfrac{3}{\sqrt{x} - \sqrt{y}}$
Rationalize the denominator: $\dfrac{h}{\sqrt{x+h} - \sqrt{x}}$
A student tries to simplify \[\dfrac{x^{-1} + y^{-1}}{4} = \dfrac{1}{4xy}\] Which exponent law was violated?
A student tries to simplify \[\dfrac{x+3}{x + 2} \cdot 4 = \dfrac{x+3\cdot 4}{x+2} = \dfrac{x+12}{x+2}\] What law or property was violated?
Show why $x = 2$ is a solution to the equation \[\dfrac{1}{x} - \dfrac{1}{x-4} = 1\]