Chapter 0: Algebra II Review

Chapter 0: Algebra II Review

Topic 1: Simplifying Polynomials & Exponential Expressions – p. 2

- Homework: Worksheet

Topic 2: Radical Expressions – p. 32

- Homework: p. 45 #33-74 Even

Topic 3: Factoring All Ways – p. 58

- Homework: p 68 & 69 #1-92 Even

Topic 4: Rational Expressions – p. 72


Topic 5: Complex Fractions – p. 80


Topic 6: Completing the Square – p. 98

- Homework: p. 106 #91-110 (Complete the Square)

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Name: _____________________________________________ Date: ____________________ Period: _________


Topic 1: Simplifying Polynomials and Exponential Expressions

Polynomial Operations:

Addition/Subtraction: Combine like-terms only

1. 2.

3. 4. Subtract from

Multiplication: Every term by every term

1. 2.

3. 4.

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Division:

1.

2.

Exponent Rules:

Remember: Exponents are always a little off from regular arithmetic rules.

Addition/Subtraction: Combine coefficients of

like-terms; exponents are unchanged

Multiplication: Multiply coefficients; add

exponents of like-bases

Division: Divide coefficients; subtract

exponents of like-bases

Negative Exponents: “I’m stuck on the wrong

side of the fraction line!” Hint: deal with these

first in complex questions!

Fractional Exponents: Power over Root

1. 2. 3.

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4. 5. 6.

7.

8. 9.

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Homework:

Perform the indicated operation.

1. (9y2 - 12y + 5) - (12y2 + 6y - 11)

2. 8(7r + y) - 3(5r - 2)

3. 2(y2 + 4y) + 6y(y - 3) 4. (8r -1) - 3(10r - 8)

5. (3g3 - 2g2 + 1)(g - 4) 6. (9 - y2)(2y + 1)

7.

8.

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Use your knowledge of exponent rules to simplify the following expressions

9. 10.

11. 12.

13. 14.

15.

16.

17.

18.

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19. 20.

21. 22.

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Name: _____________________________________________ Date: ____________________ Period: _________


Topic 2: Radical Expressions

Do Now:

Simplify the following radical expressions

1. 2.

3. 4.

Adding & Subtracting Radicals:

Just like anything else, we can only _____________________________________________________________

When adding or subtracting radicals, both the ______________________ AND the

____________________________ must be exactly the same

Before we begin to combine, we must first _________________________________________________.

Example:

Add:

1. Simplify each of the terms

2. Combine the like terms (add/subtract the

coefficients of the like-radicands)

1. 2.

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Multiplying Radicals: Multiply the numbers outside the radicals… the ________________________

Multiply the numbers inside the radicals… the _______________________

Simplify the radicals in your final answers. Do not simplify until __________________________!!!

Example:

Multiply:

1. Multiply coefficients; Multiply Radicands

2. Simplify at the end

*observe: if we simplified at the beginning,

we’d have to simplify again at the end!

1.

2.

3.

4.

3. 4.

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5.

6.

7.

8.

9.

10.

Pairs of binomials like #10 are called:

Definition: Conjugate Pairs -

_____________________________________________________________________________________

The result of multiplying conjugate pairs of radical expressions will ALWAYS be an INTEGER.

When multiplying conjugate pairs, we can skip FOIL and just multiple first & last terms. Be VERY sure you are

dealing with conjugate pairs before you take this shortcut!

Example:

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Dividing Radicals: Divide the numbers outside the radicals, divide the numbers inside the radicals.

Simplify the radicals in your final answers. If necessary, _____________________ ________

____________________

Example: Rationalizing Monomial Denominators

Divide:

1. Divide as much as possible

2. Simplify at the end – Rationalize if necessary

Example: Rationalizing Binomial Denominators

Divide:

1. Divide as much as possible (usually nothing is possible with

binomial denominators)

2. Simplify at the end – Rationalize if necessary

1.

2.

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3.

4.

5.

6.

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Name: _____________________________________________ Date: ____________________ Period: _________


Topic 3: Factoring ALL Ways

Factoring:

When we factor it is important to remember that ____________________________________________

_______________________________; we are simply rewriting it in an equivalent form.

GCF or “Greatest Common Factor” (review!):

Factoring by GCF means that we “__________________” what the terms have in _______________. This can be a

combination of numbers, variables, or both.

Factor out by GCF.

1. 2. –

3. – 4. –

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“DOTS” Factoring (review!):

Another type of factoring stems from _______________________ multiplication. This type of factoring is known as

Difference of Two Squares or “DOTS” factoring. When we factor these types of expressions, we _____________

conjugate multiplication.

Write each of the following binomials as the product of a conjugate pair.

1. – 2. –

3.

4.

Trinomial Case I Factoring (review!):

Another type of factoring is trinomial factoring. This is when we have a trinomial. In Case I factoring, the leading

coefficient is _________.

To factor these, it is helpful to look at the ______________ term and the ___________________ term.

Write each of the following trinomials in factored form.

1. 2.

3. 4.

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Factor By Grouping (review!)

We can factor by grouping when we have a polynomial that has __________________________________________.

Class Example #1: Only one variable

Steps:

1. Put terms in descending order, or with other like-factor

terms

2. Group terms in sets of 2

3. Factor each group

4. Rearrange

Use FOIL to check!

Class Example #2: More than one variable

Steps:

1. Put terms in descending order, or with other like-factor

terms

2. Group terms in sets of 2

3. Factor each group

4. Rearrange

Use FOIL to check

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Grouping Examples: 1. 2.

3. 4.

Factor Trinomials - with a leading coefficient (review!)

“Case II Factoring” To solve these examples, we use factoring by grouping by "splitting" up the middle into two factors.

Always make sure that every example is written in standard form before you try to split & factor.

Class Example #3

Steps:

1. Decide on factors & signs

2. Rewrite as 4 terms

3. Factor by grouping

4. Rearrange

Use FOIL to check!

Case II Factoring Examples: 1. 2. 5

3. 4.

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Factoring Sums & Differences of Two Cubes

General Rules

be very aware of what signs are used & when!

Worth studying and committing to memory. This will be used this year and even more in Calculus courses.

Class Example #6:

Steps:

1. Identify the cube roots of both terms

2. Plug in to the appropriate pattern (given above)

Distribute carefully to check!

Sum & Difference of Two Cubes Examples:

1. 2.

3. 4.

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Name: _____________________________________________ Date: ____________________ Period: _________


Topic 4: Rational Expressions

Define:

Rational Expression:

Domain:

Limits to domain in the real number system

Recall that the set of values which make up your domain is typically unlimited. You can plug any number you want in to

an expression or a function. (The output depends on the rule given and can vary greatly and is only limited by the rule of

the expression or function)

There is one large notable, very important, exception to the domain of rational expressions.

A denominator of a fraction can never ____________________________________!!!!!!

Examples: Determine the numbers that must be excluded from each domain:

1)

2)

3)

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Simplifying Rational Expressions

We must ensure we understand the structure of the expression we are simplifying. It is not proper to split up terms that

are joined by addition or subtraction. Rather, we can only cancel factors that appear in both the numerator and

denominator

Examples: Simplify completely by ‘factor & cancel’:

1)

2)

Review of Rational Expression Operations:

Multiplication & Division

To multiply rational expressions, factor all numerators and denominators completely. Then, across all numerator factors

and all denominator factors, cancel any matches. Multiply what remains and ensure your final answer is in simplest

form.

To divide, FIRST, perform ‘keep change reciprocal’, then proceed the same way.

Examples: Perform the indicated operation and simplify:

1)

2)

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3)

4)

Addition & Subtraction

Just like any type of adding or subtracting, we can only combine like terms. In a rational expression, like terms happen

when we have COMMON DENOMINATORS. Therefore, we must get common denominators before we can combine.

Review with arithmetic:

Least common denominator:

Find the LCD Multiply by what’s missing to get common denominators Add across

Practice the same method with a rational expression:

Steps:

1) Change subtraction if necessary.

2) Find the LCD.

3) Multiply each term by what is missing.

4) Add across.

5) Simplify if possible.

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Examples: Perform the indicated operation and simplify:

1)

2)

3)

4)

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Homework:

Simplify each rational expression:

1.

2.

3.

4.

Indicate the values for which the rational expression is undefined:

5.

6.

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Perform the indicated operation and simplify, if possible:

7.

8.

9.

10.

11.

12.

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13.

14.

15.

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Name: _____________________________________________ Date: ____________________ Period: _________


Topic 5: Complex Fractions

Simplifying Complex Fractions

Two methods for solving complex fractions are presented below. Each has a preferred time to be used, but both can be

used in any situation with careful setup. Try both options to ensure you’re fluent in both.

Class Example #1: Single rational expressions in the numerator & denominator

Rewrite left to right. Keep, change, reciprocal. Simplify.

Class Example #2: Multiple rational expressions in the numerator and/or denominator

STEPS:

1. Clean it up: Ensure ALL terms are fractions.

2. Find the least common denominator (LCD) of all of the fractions.

We do this by factoring all of the denominators of the smaller

fractions.

3. Multiply all of the fractions by the LCD we found in step 2. (By

doing this, we are multiplying the numerator & denominator of

our complex fraction by the LCD). ALL OF YOUR LITTLE

DENOMINATORS MUST CANCEL!

4. Simplify whenever possible.

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Complex Fraction Examples:

1)

2)

3)

4)

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Complex Fractions Containing Radicals

In order to simplify complex fractions with radicals, we must be able to multiply expressions with radicals

There are 2 Cases:

1. When the expression under the radical is different from the expression without the radical:

--> Here we cannot simplify further, so the expression is left as

2. When the expressions are the same inside and outside the radical:

--> Here we will follow the following steps:

1. Rewrite the radical with exponents (no radical)

2. To multiply, we will add the exponents

 3. Rewrite as a radical

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Examples:

1. 2. 3.

4. 5. 6.

Complex Fractions with Radicals

Simplify each of the following.

1.

2.

3.

4.

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Homework:

Express each complex fraction or rational expression in simplest form:

1.

2.

3.

4.

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Simplify each of the following.

5.

6.

7.

8.

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Name: _____________________________________________ Date: ____________________ Period: _________


Topic 6: Completing the Square

Completing the Square: We will force the left-side of the equation to become a perfect square trinomial.

Completing the Square 1. Move the constant term to the other side.

2. Be sure the coefficient of the highest power is one. If it is not, factor out the coefficient from

3. Create a perfect square trinomial by adding

(to both sides!) Be careful if there was a constant factored out.

4. Factor the perfect square; add the constants together.

5. Isolate the variable to solve. (Square root both sides, remove what remains)

Examples:

1. 2.

3. 4.

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5.

6.

7.

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Chapter 0: Algebra II Review