7.3 Homework: rational expressions and equations

$\frac{{\mathrm{3x}}^2+\mathrm{5x}-8}{\mathrm{6x}+\text{16}}\times \frac{{\mathrm{4x}}^3+x}{{\mathrm{3x}}^3-\mathrm{3x}}$

• A

  Simplify

• B

  What values of $x$ are not allowed in the original expression?

• C

  What values of $x$ are not allowed in your simplified expression?

$\frac{2}{{\mathrm{4x}}^3-x}+\frac{3}{{\mathrm{2x}}^3+{\mathrm{3x}}^2+x}$

• A

  Simplify

• B

  What values of $x$ are not allowed in the original expression?

• C

  What values of $x$ are not allowed in your simplified expression?

$\frac{\frac{x^2+9}{x^2-4}}{\frac{x^2+7x+12}{x^2+2x-8}}$

• A

  Simplify

• B

  What values of $x$ are not allowed in the original expression?

• C

  What values of $x$ are not allowed in your simplified expression?

• D

  Test your answer by choosing an $x$ value and plugging it into the original expression, and your simplified expression. Do they yield the same answer?

$\frac{x}{x^2-25}+$ <something> $=\frac{x^2+1}{x^3-{\mathrm{9x}}^2+\text{20}x}$

• A

  What is the something?

• B

  What values of $x$ are not allowed in the original expression?

• C

  What values of $x$ are not allowed in your simplified expression?

$\frac{x-6}{x-3}=\frac{x+\text{18}}{\mathrm{2x}+7}$

• A

  Solve for $x$ . You should get two answers.

• B

  Check by plugging one of your answers back into the original equation.

If $f\left(x\right)=\frac{1}{x}$ , find $\frac{f(x+h)-f(x)}{h}$ .

Simplify as much as possible.