Study Guide for Cht 6 & 7 Test
CHAPTER 6
Lesson 6.1- (Simplifying Rational Expressions) remember you can only cancel up & down-not top
with top or bottom with bottom.
 Any term that has a + or – next to it, is STUCK to that other term and you can’t cancel part of
it. In other words 2x + 1 can only be cancel (reduced with) another 2x + 1. You can’t
cancel any part of it individually. (look at bottom of page 358)
 If two terms are ‘exact’ opposites you need to factor out a -1 and the -1 will still be part of
your answer. Do not multiply by -1.
 Restrictions on the domain (excluded values) are any numbers that cause your
denominator to equal 0. Remember 0 on the ‘b’ottom is ‘b’ad.
 Mult/Divide rational expressions. FACTOR EVERY THING!!!! Cancel all like factors.
Remember when you divide the restrictions on the domain include both the zeros of the
numerator and denominator of the SECOND fraction.
 Leave all answers in factored form
Lesson 6.2 – ( ADD/SUBTRACTING rational expressions)
 Factor all the denominators
 To add or subtract you need to GET a common denominator, so you will need to multiply
the top and the bottom of each term but what is MISSING FROM EACH DENOMINATOR.
 Once you have a common denominator, add the like terms of all the numerators (tops)
together and put it over the common denominator
 Now factor the numerator. Reduce with the denominator if possible. Leave answers in
factored form.
 If the denominators are exact opposites, multiply the top and bottom of one of the fractions
by a -1 so denominators are now common.
 **If it is a subtraction problem, it is easiest to change the subtract to an ‘add’ and all the
terms of the following numerator to their opposite signs.
 When subtracting if denominators are exact opposites, change subtract to add and change
all the signs of the denominator instead of the numerator.
Lesson 6.3- (Complex Fractions- fractions inside fractions)
 Method 1 is easiest. Find the LCD of ALL the denominators. Multiply every term by the
complete LCD.
 If possible factor remaining terms of numerator and denominator and reduce.
Lesson 6-4- (Solving rational equations)
 Factor all denominators. List all excluded values (restrictions on the domain) before you
start.
 Find the LCD; multiply EVERY TERM by the COMPLETE LCD-(not just what is missing
from the LCD). This will cancel all denominators.
 Solve as a linear equation or a quadratic. Remember if it is a quadratic, set the equation =0
and factor to solve. DO NOT DROP OFF THE =0!!!
 Be sure you check your answers against the restrictions on the domain!
Lesson 6.5-(Application problems)
 Work Problems- remember to use _time together + _time together = # of jobs completed
time #1 alone

time #2 alone
Motion Problems-remember the rate on these problems can sometimes have two values. If
it talks about a current of a river or the head/tailwind with an airplane, you will have 2
values for the rate.

VERY IMPORTANT- The speed of the actual object goes first and the speed of the
wind/current goes second. Example: speed of boat ± speed of the current.
Lesson 6.6- (Division of Polynomials)
 If you divide by a monomial, you can divide each term individually.
 If you divide by a binomial/trinomial, you will need to use long division
 With long division, you need to make sure the polynomial is in descending order. You also
need to use 0’s as place value holders.
 You keep going until the remainder is a less degree than what you are dividing by.
Lesson 6.8 – (Formulas, Applications and Variation)
 When the equation is written in mostly variables remember you are going to ‘undo’ them
(move them to the opposite side of the equation) by just using the inverse operation. If you
are multiplying by G, then to move it you are going to divide by G.
 If the variable you are trying to solve for is in more than one term you need to get all of
those on the same side and then factor the variable you are solving for out of all terms.
 Variation- directly means multiply, proportionally means multiply, inversely means divide,
jointly means multiply by more than one thing at a time.
 Solving a variation problem- 1st- write the equation, 2nd-use the first set of given info to
solve for k, 3rd- use what you found for k and the 2nd set of info to find the missing value.
CHAPTER 7
Lesson 7.1 – (Radical expressions and functions)
 √𝑎2 = |𝑎|- even, steven, odd. (Index is even, power under radical is even and solution
power is odd-you need to use absolute value bars. That’s the only time.
Lesson 7.2- (rational exponents)
𝑛
 √𝑎 = a1/n (the index becomes the denominator)
 a –m/n = 1/ a m/n- any time you have a negative power you move it to the opposite side of the
fraction and make the power positive (even if the power is a fraction)
 If you need to simplify radical expressions that don’t have the same indexes, change them
to rational exponents and get a common denominator.
Lesson 7.3 – (multiplying radical expressions)
 Think outside with outside/inside with inside
 Can only multiply if indexes are the same, if not change to rational form and get a common
denominator
 Take things out from under the radical in groups that are as big as the index
Lesson 7.4 – (dividing radical expressions)
 If radical in denominator you will need to rationalize the denominator. If the radical in
denominator is a sq root then just multiply the top and bottom by the exact same radical.
If the index is anything other than a 2, you will need to multiply the top and bottom of the
fraction by ‘what is missing’ from the radical in the denominator.
 If radical in the denominator is part of a binomial you will need to multiply the top and
bottom by the conjugate of the denominator Example: 1-√3 has conjugate of 1+√3.
Lesson 7.5 – (Expression containing several radical terms)
 You can only add like radicals.
 Multiplying/dividing radicals with different indices, you will need to change to rational
(fractional form) and get a common denominator.
Lesson 7.6- (Solving radical equations)
 Get the radical on a side by itself before you start. If more than one radical, put them each
on a separate side.
 If there is more than one variable remember you have to go through the complete process
twice.
 Remember (2 + √3)2 ≠ 4 + 3
Lesson 7.7- (Distance Formula, Midpoint, and other Applications)
 Use a2 + b2 = c2 for any right triangle. Remember c is always the long side which is the
hypotenuse.
 Don’t forget if you have to put the square root into the problem you have to put ± so you
include both cases.
 Formulas for the distance, the midpoint, and the length of sides of a the special right
triangles are on the posters at the front of my room (just don’t tell anyone lol)
Lesson 7-8 – (Complex Numbers)
 Remember that i2 = -1 and √−1 = i
 Complex numbers should always be written with the imaginary last a ± bi
 The conjugate of i = -i, and the conjugate of 2 – 3i = 2 + 3i
 If the power on i is greater than 4, then divide the power by 4 to figure out the remainder.