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

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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!
  • 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

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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.
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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

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  • 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
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