Algebra 2 Review - Mr. Scott's Webpage · PDF fileAlgebra 2 Review Date:_____ 1 Algebra 2 Review The following topics are a ... Trinomials—One way to factor trinomials (ax2 bx c)

PreCalculus 300 Name:_______________________________

Algebra 2 Review Date:_______________________________

1

Algebra 2 Review

The following topics are a review of some of what you learned last year in Algebra 2. I will spend

some time reviewing them in class. You are responsible for knowing these topics and coming to

see me for any extra help you may need.

Factoring

1. GCF (Greatest Common Factor)— The GCF is always the first thing you look for when

factoring. There may be additional factoring you will need to do after the GCF. Even if there isn’t

a GCF, you may still need to factor the polynomial. Do not assume that if there isn’t a GCF the

polynomial is prime. Make sure you check your answer by distributing.

4232 216 zxyzyx 23 39 xx xzzyyx 869 43

2. Difference of Squares (DOS)—In order to have DOS, you must have exactly 2 terms, both of

which are perfect squares. All exponents of variables must be even. (Even though 9 is a perfect

square, 9x isn’t.) You also have to have subtraction. There is no such factoring as sum of squares.

The way to factor DOS is ).)((22 bababa

42x

44 yx

zzx 1923 2

252c

2425 d

9)7( 2x

3. Trinomials—One way to factor trinomials )( 2 cbxax is by using the sum-product rule. When

,1a you find 2 numbers that multiply to the c value and add to the b value. Don’t forget to look

for a GCF first!

1452 xx

2222 3093 yzxyzyzx

2422 2 xx

If you do have an a term other than 1, you can either guess-and-check or split the middle term.

2076 2 xx

1. Multiply a and c.

2. Find 2 numbers that multiply to the number you got in step #1 and add to b.

3. Split the middle term and rewrite the trinomial according to the 2 numbers you came up

with in step #2. If one of the numbers in step #2 is positive and the other is negative, it’s

better to write the negative term before the positive one.



2

4. Group the first 2 terms and the last 2 terms with parentheses.

5. Factor each set of parentheses using GCF. Whatever remains after taking the GCF should

be the same.

6. Regroup by taking each GCF in 1 set of parentheses and whatever remains in the other

set.

7. Check your work by FOILing!

15164 2 xx 62220 2 xx

4. Sum/Difference of Cubes (S/DOC)— In order to have S/DOC, you must have exactly 2 terms,

both of which are perfect cubes. All exponents of variables must be multiples of 3. (Even though 8

is a perfect cube, 8x isn’t.) Unlike DOS, you may have addition or subtraction. The way to factor

S/DOC is 3 3 2 2( )( )a b a b a ab b or 3 3 2 2( )( ).a b a b a ab b If you factor correctly,

the quadratic (trinomial) part of S/DOC factoring should not be factorable. Look for a GCF first!

1253x 3436x 6 32 38 432x x

Here are some additional practice problems.

513 164 zz 8103 2 yy 234 16182 xxx

zzx 1355 3

232 4x 1008016 2 xx

zzy 1082 6

22 1514730 baba 25449 x

Completing the Square (CTS)

CTS is used to take a quadratic equation )( 2 cbxaxy and make it look like .)( 2hxaky

Take a look at .262 xxy 1. Make sure .1a

2. Send the c term to the opposite side of the equation.

3. Take the b term, divide it by 2, and then square the quotient. This number will be added

to both sides of the equation in order to make a perfect square trinomial (PST) on one side.

4. Factor the side of the equation with the PST.

5. Simplify the other side of the equation to make it look like .ky



3

7102 xxy 1142 xxy 2 3 10y x x

Take a look at .116363 2 xxy Here, .1a

1. Send the c term to the opposite side of the equation.

2. Since ,1a we need to make it 1 by taking out the GCF.

3. Take the b term, divide it by 2, and then square the quotient. This number will be added

to both sides of the equation in order to make a perfect square trinomial (PST) on one side.

When adding this number to the other side of the equation, multiply it by the GCF.

4. Factor the side of the equation with the PST. If you do it correctly, it should look like

.)( 2hx

5. Simplify the other side of the equation to make it look like .ky

242644 2 xxy 5305 2 xxy 1073 2 xxy

11243 2 xxy 22 20 13y x x

22 2 9y x x



4

Rational Exponents and Exponent Laws

Let’s first recall the properties…

Product Property: nmnm aaa

Example: 2 3 2 3 5x x x x

Quotient Property: nm

n

m

aa

a where 0a

Example:

55 2 3

2

xx x

x

Power of a Power Property: mnnm aa

Example: 5

2 2 5 10x x x

Power of a Quotient Property: m

mm

b

a

b

a

where 0b

Example:

3 3

3

x x

y y

Power of a Product Property: mmmbaab

Example: 1 11

22 2 2264 64 8x x x

Example: 4

2 3 2 4 3 4 4 8 12 4x y z x y z x y z

Negative Exponents: n

n

bb

1

and n

nb

b

1

Example: 2

2

1 13

3 9

and 2

2

13 9

3

Zero Exponents: 0 1a

Example:

010 1

Definition of Rational Exponents: n

m

b n mb mn b

Example: 2

23 2 2338 8 8 2 4



5

Now you practice!! If you need extra review try visiting…

http://www.khanacademy.org/math/algebra/exponents-radicals/v/rational-exponents-and-exponent-

laws

Rewrite in radical form and simplify.

1.

1

5x 2. 2

5y

Rewrite in exponential form and simplify. Use the smallest base possible.

3. 1.2y 4. 37x 5. 3

7x

6. 2

3 a 7. 4 36 8. 6

3 5xy

Simplify completely.

9.

74

7x

10.

3

272 11.

3

4625

12. 3

481 13. 1

15 532y 14.

1 1

4 43 27

http://www.khanacademy.org/math/algebra/exponents-radicals/v/rational-exponents-and-exponent-laws

http://www.khanacademy.org/math/algebra/exponents-radicals/v/rational-exponents-and-exponent-laws



6

15.

21

0.253236 125 81 16.

522

3

3 273

1yxxyyx

17.

3

4 2 29x y

18. 23423 96 bbaba 19.

13 3

9

x

x

20.

122 1

3 6x y

21.

152

3

1

3

x

y

22.

16 31

2x

23.

0.514

18

16

81

x

y

24.

34 8 4

12

81

16

x y

x

25. 4 9 2 103 64w x y z 26. 1212

22

27. 233

243

6

6



7

Radicals

Simplifying Radicals: We always want the smallest possible number under the radical, otherwise

known as the radicand. Look for factors that are Perfect Squares, Perfect Cubes, Perfect Fourths,

etc. The types of factors you look for need to match the index of the radical.

1. 3 554x 2. 492 722 zyzy

Multiplying Radicals

Example: 2329186363

Multiply everything under the radical together and then simplify!

Example: 726123612364318423

Multiply everything outside of the radical together and everything under the radical together

and then simply!

3. 2 4 211 44x y x y 4. )618( 432 yxyx )18

3

1( 32 yx

Adding & Subtracting Radicals:

You can only add and subtract radicals when they are “like radicals.”

How do you define “like radicals?”

5. 1085484

127

3

2 6. z9 zz 424

3 8 3 881z



8

Dividing Radicals: Radicals are not allowed in the Denominator: “Rationalize” it! Multiplying a fraction by a convenient form of “1.”

** When rationalizing the denominator, multiply the numerator and denominator

by what will make it a perfect root!

7. 3

2

3 8.

1

4x 3

232

3

x

Fractions with complex denominators

Conjugates: We can create a conjugate of any binomial.

61

274

32

With a binomial in the denominator, multiply the fraction by a form of 1 that uses the conjugate.

***** When you multiply by the conjugate, the radical goes away!

9. 32

2

10.

36

72

Additional resource:

http://www.khanacademy.org/math/algebra/exponents-radicals/v/simplifying-radical-expressions1

http://www.khanacademy.org/math/algebra/exponents-radicals/v/simplifying-radical-expressions1



9

Now it is your turn!

Simplify the following; express all answers in lowest terms with no negative exponents.

1.

856

2

134 2. 3

3

2

x

3. 32

35

4. y

2

3 yy 3243 5 yy

4

7813 5 3 5192 y

5. 822

83

6.

3

3

4

32

7. 3 2

3 5

3

162

x

x 8.

xy

x

5

3

9. 3

4

7x 23 5

2

5320 xx

73 8 45240 xx



10

Radical Equations

Simple Radical Equation:

http://www.youtube.com/watch?v=y4C81qAa3pY&safety_mode=true&persist_safety_mode=1&safe=activ

e

Complex Radical Equation:

http://www.youtube.com/watch?v=g6nGcnVB8BM&safety_mode=true&persist_safety_mode=1&safe=acti

ve

There are five steps to solve a radical equation.

1. Isolate the radical

2. Make sure the radical equals a positive number

3. Raise each side to a power to get rid of the

radical

4. Solve

5. Check!

For equations with fractional exponents:

1. Isolate the fractional exponent

2. Raise to a power that will cancel the fractional

exponent

3. Simplify and repeat step 2 if necessary

4. Solve

5. Check!

1. 45 c 2. 312 p

3. 636 m 4. 4306 m

5. 41032 k 6. 127 ww

http://www.youtube.com/watch?v=y4C81qAa3pY&safety_mode=true&persist_safety_mode=1&safe=active

http://www.youtube.com/watch?v=y4C81qAa3pY&safety_mode=true&persist_safety_mode=1&safe=active

http://www.youtube.com/watch?v=g6nGcnVB8BM&safety_mode=true&persist_safety_mode=1&safe=active

http://www.youtube.com/watch?v=g6nGcnVB8BM&safety_mode=true&persist_safety_mode=1&safe=active



11

7. 6151 xx 8. 4322 mm

9. 0255210 xx 10. 2

1

2

1

23823

xx

11. 6

1

3

1

112 xx 12. 364123

4

x



12

Rational Root Theorem

Factor Theorem: If x a is a factor of f x , then 0f a .

Rational Root Theorem: If a rational number p

q is the root of a polynomial equation,

0...21 zyxcxbxax nnn , then p is a factor of the constant (z) and q is a factor of the

leading coefficient (a).

The purpose of the Rational Root Theorem is to determine the possible roots of a higher degree

function.

Example: 3 23 4 47 34f x x x x

a) Find all possible roots using the Rational Root Theorem.

b) Use your calculator to find the actual roots (roots that make the equation equal 0)

c) Now use synthetic division and other factoring methods to break down the polynomial into

smaller factors and find ALL exact roots. (You may need to use the Quadratic Formula at the

end.)

For further explanation, watch this video:

http://www.youtube.com/watch?v=xJvrhlqwCr0&feature=related&safety_mode=true&persist_safet

y_mode=1&safe=active

http://www.youtube.com/watch?v=xJvrhlqwCr0&feature=related&safety_mode=true&persist_safety_mode=1&safe=active

http://www.youtube.com/watch?v=xJvrhlqwCr0&feature=related&safety_mode=true&persist_safety_mode=1&safe=active



13

Practice: Find all exact roots (real and imaginary).

1. 62534 23 xxxy 2. 355 23 xxxy

3. 362753 234 xxxxy 4. 424 81y x x

5. 1692 23 xxxy 6. 8823 xxxy

7. 3216946 234 xxxxy 8. 4 29 117 108y x x



14

Solving Rational Equations

Recall that a rational expression is a fraction where the numerator and/or denominator is a

polynomial.

To simplify a rational expression:

Factor the numerator completely

Factor the denominator completely

Cancel any common factors

Example: ab

abba

5

1515 22 =

Recall that when multiplying two fractions, we simplify and then multiply across, numerator x

numerator, denominator x denominator. Same idea when multiplying rational expressions.

Example: aa

a

a

a

2

1

1

422

2

When dividing two fractions, we multiply by the reciprocal of the second fraction. Same idea when

dividing rational expressions.

Example: 84

246

42

123

y

y

y

y

Recall that when adding fractions, we find a common denominator first, rewrite the fractions with

the common denominator, then add the numerators. Same idea when adding rational expressions.

1. Factor the denominators

2. Determine the Least Common Denominator (LCD)

3. Multiply in the missing pieces of the LCD

4. Simplify the numerators

5. Add numerators

6. Check to see if numerator is factorable, then simplify

Example: 33

5

45

12

x

x

xx



15

When subtracting rational expressions, the only difference is in Step #5 above, where you subtract

numerators instead of adding them!

Example: 633

12

693 22

xx

x

xx

x

We use the techniques above when solving rational equations. Two types:

1. One rational expression is equal to another rational expression.

Method of solving: CROSS MULTIPLY

2. Sum or Difference of rational expressions is equal to another rational expression.

Method of solving: MULTIPLY EVERYTHING BY THE LCD! (LCM)

***Always remember to check your solutions

For further clarification: http://www.khanacademy.org/math/algebra/ratios-proportions/v/solving-

rational-equations-3

Solve and check for extraneous solutions.

1.1

15

22

52

xx

Check:

http://www.khanacademy.org/math/algebra/ratios-proportions/v/solving-rational-equations-3

http://www.khanacademy.org/math/algebra/ratios-proportions/v/solving-rational-equations-3



16

2. 2

1

5

2

2

1

xx Check:

3. 1

2

1

1

1

12

xxx

Check:

4.25

4

5

2

5

32

xxx Check:

5. 221

k

k

k

k Check:

6.xx

x

xx

xx 3

3

6

3

932

2

Check:

7.4

4

22

22

2

x

x

x

x

x Check:



17

8.12

22

4

3

3

22

xx

x

xx Check:

9. Check:

10. Check:

11.2

2

2

xx

xx Check:

12. 1

3

1

2

x

x

x

xx

Check:

2 2

4 4

2 1 2 2 1

x

x x x x

3 3 9 6 3

( 3) 3

x x x

x x x x

Documents

Algebra 2 Review - Mr. Scott's Webpage · PDF fileAlgebra 2 Review Date:_____ 1 Algebra 2 Review The following topics are a ... Trinomials—One way to factor trinomials (ax2 bx c)