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Accuplacer Review Workshop

Elementary Algebra Part II

Week Three

Preparing for the Algebra Test: To Place into College Level Math

It is important to know that the accuplacer assesses three areas of Algebra:

! Numerical Skills/Pre-Algebra ! Elementary Algebra ! Intermediate Algebra

These areas are required prerequisites for College Algebra.

The following workshop will hopefully aide you in your review of Elementary Algebra.

NOTE: Although some questions are “basic skills” many accuplacer questions require you to put several math concepts together, or work multiple step problems to determine the final answer.

Includes internet links to instructional videos for additional resources:

http://www.mathispower4u.com (Arithmetic Video Library)

http://www.purplemath.com

http://www.coolmath.com

http://www.testprepreview.com

http://www.kahnacademy.org

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Elementary Algebra Part II:

Questions in the Elementary Algebra range from introductory algebra concepts and skills to the knowledge and skills considered necessary to enter an intermediate algebra course. The required skills also cover a majority of items from these following categories:

• Simplifying Radicals (Using multiplication and division properties)

• Basic Operations of Radicals 1. Addition 2. Subtraction 3. Multiplication

• Basic Operation of Polynomials 1. Addition 2. Subtraction 3. Multiplication 4. Division

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Simplifying Radicals: To simplify a radical means to make the number inside the radical as small as possible (but still a whole number).

This is done by using the multiplication or division properties of radicals.

Example 1: Use the product rule to simplify 8

8 = (4)(2) = 4 2 = 2 2

Example 2: Use the product rule to multiply 3 ! • 2 !

3! • 2! = (3)(2)! = 6!

Since we cannot take the cube root of 6 and 6 does not have any factors we can take the cube root of, this is the simplified answer.

Example 3: Use the product rule to multiply 2𝑥²y! • 5𝑥𝑦!

2𝑥²𝑦! • 5𝑥𝑦! = (2𝑥!𝑦)(5𝑥𝑦)! = 10𝑥³𝑦²!

Since we cannot take the fourth root of any of the variables inside the radical sign and 10 does not have any factors with a fourth root this is the simplified answer.

Multiplication Property

A Product of Two Radicals with the same Index Number

√𝒙! • !y𝒏 = !𝑥𝑦!

In other words, when you are multiplying two radicals they must have the same index number. Then you can write the product under one radical with the common index.

Note: This rule can be used in either direction

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Example 4: Use the quotient rule to simplify!!"

!!"

= !!"

= !!

Since we cannot take the square root of 2 and 2 does not have any factors that we can take the square root of, this is the simplified answer.

Example 5: Use the quotient rule to simply !!

!

!!

! =

!!

!! = !!

!

Since we cannot take the cube root of 5 and 5 does not have any factors that we can take the cube root of, this is the simplified answer.

Example 6: Use the quotient rule to divide then simplify !"!

!"!= !"

!= 10

Since we cannot take the square root of 10 and 10 does not have any factors we can take the square root of, this is the simplified answer.

Quotient Property

A quotient of Two Radicals with the same Index Number

If n is even, x and y represent any nonnegative real number and y does not

equal zero.

If n is odd, x and y represent any real number and y does not equal zero.

√! !

√!! = !!!

!

Note: This rule can also work in either direction.

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Basic Operations of Radicals:

Example 7: Add 2 20𝑥 + 3 5𝑥

First Simplify: 2 (4)(5𝑥) + 3 5𝑥

2 4 5𝑥 + 3 5𝑥

2 (2) 5𝑥 + 3 5𝑥

4 5𝑥 + 3 5𝑥

Collect like radicals 7 5𝑥

Example 8: Combine like radicals 3𝑏³! - 3b 24! + 2 81𝑏³!

First Simplify: b 3! – 6b 3! + 6b 3!

Collect like radicals b 3!

Like Radicals

Like radicals are radicals that have the same root number AND radicand (expression under the radical sign).

Example: !𝑥𝑦 and 5!𝑥𝑦

-2 √4𝑎𝑏 ! and 7√4𝑎𝑏!

Adding and Subtracting Radical Expressions

Step 1: Simplify the radicals

Step 2: Combine like radicals

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Example 9: Add the radicals !"!!

! +

! !!

!

First Simplify: !" ! !!

! +

! !!

!

! !!

! +

! !!

!

Combine like radicals ! !!

! = 𝑥!

Example 10: Multiply and Simplify 2 ( 3 - 8)

Step 11: Multiply the radical expression AND Step 2: Simplify

2 ( 3 - 8) = 6 - 16 = 6 – 4

Example 11: Multiply and Simplify ( 𝑎 – 5) (3 𝑎 + 7)

Step 12: Multiply the radical expression AND Step 2: Simplify

( 𝑎 - 5) (3 𝑎 + 7) = 3 𝑎² + 7 𝑎 - 15 𝑎 – 35 = 3a - 8 𝑎 – 35

Multiply Radical Expressions

Step 1: Multiply the radical expression

Step 2: Simplify the radicals

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Polynomials: A polynomial is a finite sum of terms separated by “+” and “-“ signs

and has constants, variables and exponents 0,1,2,3…. but it never

has division by a variable. Example: 3𝑥! - 5𝑥! + x - 10

Monomial, Binomial, Trinomial

These are special names for polynomials with 1, 2, or 3 terms:

Example: 3xy² is a monomial (1 term)

5x – 1 is a binomial (2 terms)

3x + 5y² = 3 is a trinomial (3 terms)

Combining like Terms

Like Terms are terms that have the exact same variables raised to the exact same exponents

Example: 3𝑥! - 5𝑥!

Simplify by combining like terms: 5𝑥! + 7x - 2𝑥! -10x + 5 =

3𝑥! – 3x + 5

Degree of the Polynomial

The degree of the polynomial is the largest degree of all its terms

Example: -4𝑥! + 7𝑥! -3

The polynomial has a degree of 4

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Basic Operations of Polynomial:

Example 12: Perform the indicated operation and simplify

(5𝑥! – 4x + 10) + (3𝑥! – 2x -12) = 5𝑥! -4x + 10 + 3𝑥! -2x -12 =

8𝑥! – 6x + 2

Example 13: Perform the indicated operation and simplify

(5x – 2y + 1) – (2x – 7y + 4) = 5x – 2y + 1 – 2x + 7y – 4 =

3x + 5y – 3

Adding Polynomials

Step 1: Remove the parenthesis

If there is only a + sign in front of ( ), this is multiplication by positive one so all the terms inside of the ( ) remain the same when you remove the parenthesis

Step 2: Combine like terms

Subtracting Polynomials

Step 1: Remove the parenthesis

If there is a minus sign in front of the ( ) then distribute it by multiplying every term in the ( ) by a negative one

Step 2: Combine like terms

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Example 14: Multiply (-7𝑥!) (5𝑥!) = (-7)(5) (𝑥!𝑥!)

= -35𝑥!

Example 15: Multiply -2(5ab + 3𝑎!𝑏! + 7𝑎!𝑏!)

= 10𝑎!b - 6𝑎!𝑏! - 14𝑎!𝑏!

Example 16: Multiply (3x + 5) (2x – 7)

= 6𝑥! – 21x + 10x – 35

= 6𝑥! – 11x – 35

Example 17: Multiply (3y -1) (2𝑦! + 5y -8)

= 6𝑦! + 15𝑦! – 24y - 2𝑦! -5y + 8

= 6𝑦! + 13𝑦! – 29y + 8

Multiplying Polynomials

When multiplying two polynomials together, use the distributive property on the first polynomial until every term in it has been multiplied times every term in the other polynomial

(Monomial) (Monomial)

(Monomial) (Polynomial)

(Binomial) (Binomial)

(Polynomial) (Polynomial)

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Example 18: Divide !!!!!!!!!"

!!

= !!! !!

+ !!! !!

- !"!!

= !!!

! + 3𝑥! -

!"!

= !!𝑥! + 3𝑥! -

!"!

Dividing A Polynomial by A Monomial

Step 1: Use the distributive property to write every term of the numerator over the monomial in the denominator.

Step 2: Simplify the fractions