Download Chapter “-1” Study Guide- Key Factoring The purpose of factoring is

Chapter “-1” Study Guide- Key

Factoring
The purpose of factoring is to write a polynomial as the product of two or more other
polynomials. When factoring polynomials always look for special patterns. The special
patterns are:
1)
2)
GMF- Greatest Monomial Factor
GBF- Greatest Binomial Factor
3)
DOTS- Difference of Two Squares
a 2  b 2  (a  b)(a  b)
4)
PST- Perfect Square Trinomial
a 2  2ab  b 2  (a  b) 2 OR
a 2  2ab  b 2  (a  b) 2
5)
GT- General Trinomial
Two other special patterns to look for are:

(a 3  b 3 )  (a  b)(a 2  ab  b 2 )
1)
SOTC- Sum of two cubes
2)
DOTC- Difference of two cubes (a 3  b 3 )  (a  b)(a 2  ab  b 2 )
Long Division/Synthetic Division
In order to determine if a polynomial is a factor of another polynomial, you can perform
long division or synthetic division. If the remainder is zero, it’s a factor.
There are two conditions on using synthetic division.
1)
2)
The polynomial must be in standard form (exponents in descending order).
The divisor must be in the form (x – r) where r is a number/constant.

Polynomials/Rules of Exponents
Monomial = __one term i.e. x2
Binomial = ___two terms i.e. x2 + 4x
Trinomial = __three terms i.e. x2 + 3x + 2
Polynomial = _many terms i.e. x3 + 4x2 – 3xy + 4y____
___________
When adding and/or subtracting polynomials, add like terms. Like terms are considered
terms that have the same “variable part”. DO NOT ADD OR SUBTRACT EXPONENTS- leave
them the same.
When multiplying polynomials, ____Multiply as you would normally, adding the exponents!
Remember FOIL when you multiply a binomial by a binomial! When multiplying a monomial
and a polynomial, distribute! When multiplying a polynomial and a polynomial be sure to
multiply each term in one polynomial by each term in the other polynomial.
When dividing polynomials, Divide each term in the numerator by the term (monomial) in
the denominator. Remember the rules of exponents (subtract them!). Polynomial divided
by a binomial use long division or synthetic division (see notes above for rules!)
Rules of exponents:
a m n
Example:
a mn
Example:
a mn
Example:
4) a  n 
1
an
Example:
1

an
an
Example:
1
Example:
1)
am  an 
2) (a m )n 
3)
5)
am

an
6) a 0 

Radicals
When simplifying radicals, think “perfect squares”. To add and subtract radicals, the
radicands must be the same! To multiply radicals, multiply parts not under the radical
together and the parts under the radical together. When dividing radicals, the main rule is
there CANNOT be a radical in the denominator. If there is, multiply the top and bottom by
the radical or the conjugate of the denominator. This is sometimes called, “rationalizing the
denominator”.

Radical Equations
When solving radical equations, the following steps must take place:
1) Isolate the variable part. *If the variable you are solving for is under the radical.
2) Square both sides in order to get rid of the radical. If it’s a cube root, cube both sides. If
it’s a fourth root, raise both sides to the 4th power.
3) Solve for the variable.
4) Check for extraneous solutions.

Rational Exponents
Rational exponents are exponents which are fractions.
1
2
a 
a
x
x
1
2
3
v
v
1
3
2
3
j  3 j2
When simplifying rational exponents be sure to convert from exponent form to radical form
and vice versa. This will insure you are completely simplified!

Complex Numbers
A complex number is also known as an imaginary number. The standard form of a
complex number is: a + bi , where “a” is the real part and “bi” is the imaginary part.
i
1
i 2  -1
When adding complex numbers, you add the real part with the real part and the
imaginary part with the imaginary part.
When multiplying complex numbers, FOIL!
When dividing complex numbers, remember there CANNOT be an imaginary number in
the denominator. You must multiply top and bottom by the imaginary number or the
complex conjugate of the denominator.

Solving quadratic equations
There are four ways to solve quadratic equations. They are:
1) Graphing
2) Factoring
3) Completing the Square
4) Quadratic Formula
Two of these four ways ALWAYS WORK! They are: Completing the Square and Quadratic
Formula.
Completing the Square steps:
1) Set equation equal to the constant term (CT).
2) If the leading coefficient of the QT is different than a “1”, divide the QT and the LT by it.
3) Take ½ of your LT and square it.
4) Add the result of step #3 to both sides of your equation. If you divided out a number
from step #2 be sure to add the product of that number and the number you added.
5) Write one side as a PST and combine like terms on the other side.
6) Take the square root of both sides. Don’t forget your +/-!
7) Solve for your variable. Remember- you should always have two answers!
Quadratic Formula is:
x
 b  b 2  4ac
2a

Rational expressions
A rational expression is a fraction. In order to simplify rational expressions, you must write
both your numerator and denominator as the product of two or more polynomials. Since
each are written as products, you can cancel like terms.
Multiplying rational expressions involve writing each numerator and denominator as a
product of two or more polynomials and canceling out like terms within each fraction and
“criss-cross”. Then multiply numerators and multiply denominators.
Dividing rational expressions involve the same steps as multiplying however two extra steps
are required. The two steps are: 1) change division to multiplication and 2) invert the
second fraction (“flip it”).
The most important thing to remember when adding and/or subtracting rational
expressions is to have a common denominator. It may be helpful to write the LCD off to the
side. Multiply each fraction by the “piece” of the LCD that’s missing.

Complex Fractions
A complex fraction is a “fraction within a fraction”. To simplify complex fractions, first find
the LCD of all denominators. Then, multiply each fraction by the LCD and eliminate
denominators. Be sure to simplify your final answer if applicable.

Other
The bonus question work may be a good idea to put here!