Name: ______________________________
Mr. Art
Date: _______________
Period: ______________
Factoring, Solving Quadratic Equations Algebraically, Algebraic Functions
I. Factoring = dividing an expression into factors that if multiplied together will result in the original expression
1) GCF = Greatest Common Factor
a) GCF - what is in common between all terms
1) coefficients (numbers) the largest number that evenly goes into all original numbers
2) variables same variable, smallest exponent
b) Pull the GCF out (outside the parentheses)
c) Leave the "leftovers" inside parentheses (by dividing each term by the GCF)
d) Check your answer by distributing the GCF by the expression inside the parentheses
or
*After factoring using the GCF method, the number of terms inside the parentheses should be the same as
the number of terms in the original expression.
*If you factor out a GCF that is identical to one of the terms in the original expression, don’t forget to
leave 1 in its place inside the parentheses.
2) DOTS = Difference of Two Perfect Squares
* Must have a binomial where the terms are separated by subtraction AND each term is a perfect square.
a) Double Parentheses
b) Plus, Minus or Minus, Plus (either is okay, as long as they are different)
c) The square root of the first term and the square root of the second.
*Remember, you take the square root when factoring with the DOTS method, you don’t divide by 2!
3) A&M = Trinomial (Add & Multiply)
* When you can't take out a GCF (or you already have taken one out) and you have a trinomial
a) Make sure the trinomial is in degree order (exponents go from 2, 1, 0)
ax2 + bx + c
b) Write an "A" above the coefficient of the middle (x) term & "M" above last term (no variable)
c) Find the two numbers that will Add to the middle term's coefficient and Multiply to the third term
*Tip: Find the M first, then the A second
y2 + y - 20
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II. Factoring Completely = factor multiple times until you can't anymore
196 - 4y2
4x2 - 24x - 28
GCF
or
GCF
DOTS
A&M
III. Solving Quadratic Equations Algebraically
1) Regular Quadratic Equations
ax2 + bx + c = 0
a) Set equation equal to zero
b) Factor completely
c) Set each factor equal to zero ("T" it off)
d) Solve each equation for the variable
e) Solution Set (in braces)
may have 2 solution (most common), 1 solution or no solutions
GCF
x2 - 6x = 0
DOTS
y2 = 64
A&M
y2 - y - 30 = 0
Factoring Completely
4x2 - 144 = 0
{ -6, 6 }
2) Proportions that lead to Quadratic Equations
a) Immediately put parentheses around binomials & trinomials in fractions
b) Cross Multiply
c) Distribute where necessary
x 144
x2 x2


9
x
2x  1
3
distribute
2
3) Linear-Quadratic Systems Algebraically
a) Substitution Method
Original problem:
1) Substitute the expression equal to
one of the variables into the second
equation for that variable
2) Set the quadratic equation = 0
3) Solve algebraically
b) Set Equal
* If you can set each equation equal to the same thing, then you can set them equal to each other
* Now that both equations equal y, you can set them equal to each other!
Original problem:
1) Set = y
2) Set them = to each other
3) Choose your solution
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IV. Algebraic Fractions
I. Undefined Fractions
-
1)
2)
II. Simplifying Algebraic Fractions
-
III. Multiplying Algebraic Fractions
-
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IV. Dividing Algebraic Fractions
- To divide algebraic fractions, you must multiply the 1st fraction by the reciprocal (flip) of the 2nd.
V. Adding Algebraic Fractions
- To add algebraic fractions you must get common denominators
V. Subtracting Algebraic Fractions
- To subtracting algebraic fractions you must get common denominators, then distribute when necessary
1)
2) What is
2x 2
12 x

in simplest form?
x6 x6
5