Problems and tutorials from Unit 1

## Basic FactoringEdit

- Simplification
- Take out GCF
- Completing the square
- Quadratic Formula
- Sum/Difference of Perfect Cubes*
- Remember SOAP (Same, Opposite, Always Positive)
- Sum: (a + b)(a² - ab + b²)
- Difference: (a - b)(a² + ab + b²)

- Grouping

## Heart ProblemsEdit

- Identify the heart piece (the one with a difference variable than the other pieces)

ex: x² + 4x + 4 - 9y²

- The "Heart Piece" is -9y² because it is the only y.
- The piece alone factors into -3y and 3y. You add these numbers into the factors of the polynomial (in this case, it's x² + 4x + 4) which factors to (x-2)(x-2)
- Final solution is (x - 2 + 3y)(x - 2 - 3y)

Sample problem

## Factoring with Fraction ExponentsEdit

- Take out a GCF -- Take out whichever exponent is smaller. Then, factor.
- Remember that an exponent times an exponent is you add the exponents together.
**Don't multiply them!**

- Remember that an exponent times an exponent is you add the exponents together.
- If you are left with a negative exponent, you have to simplify.
- Just switch the piece with a negative exponent from the numerator to the denominator, and switch the sign of the exponent (ex. -1/4 would become 1/4)

Sample problem

## Complex FractionsEdit

- Turn every piece in the numerator and denominator into a fraction.
- Make all fractions have a common demonimator through multiplication. Make sure you multiply the numerator by whatever you are multiplying the denominator in order to get the common.
- Once your common denom, take the denominators away completely.
- Simplify and factor further if necessary.
- Be careful with ones that have x and h as variables.

## Restricting DomainEdit

- Check all fractions used in the problem, during
**all**stages of your work, and figure out which values of x or h (or another variable) make the denominator 0. - Look at the previous problem for example
- If there is a piece under a radical, than the variable must be equal to or greater than 0 (can't have a negative under a root)
- Write points as coordinates. Use [] if including a value, () if not including a value.

Sample problem

## InvertingEdit

- To find the inverse of a function, swap x and y and solve for y.
- A function can only have an inverse if it passes the horizontal line test.
- If a function doesn't, you have to restrict it to a domain it that does.
- For example, if you have a parabola, cut it in half down the middle and give the domain of the left
**OR**the right side. See sample problem

- For example, if you have a parabola, cut it in half down the middle and give the domain of the left

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