College Algebra

Information Systems Society - Academics Committee Institute of Information and Computing Studies

University of Santo Tomas

By: J.G.M. Manuel

MATH 101 (College Algebra)

Reviewer for Preliminary Examinations

MODULE 1

Real Numbers and their properties

Real Numbers

Rational Numbers – numbers which can be expressed as a ratio of 2 integers.

Irrational Numbers – numbers which cannot be expressed as a ratio of 2 integers.

Integers – consists of the set of whole numbers and their negative.

Non-integers – Fractions, Non-terminating repeating Decimals, Terminating Decimals

Whole Numbers – Counting numbers and Zero

Natural Numbers – Counting numbers ONLY

Properties of Real Numbers

Commutative Property – you can add numbers in any order (Addition) Ex: 3a+2=2+3a

- You can multiply numbers in any order (Multiplication) Ex:

(3)(4)(6a)=(6a)(4)(3)

Associative Property – you can group numbers in a sum in anyway you want and still get the

same answer. (Addition) Ex.: (3x+2x)+9=3x+(2x+9)

- You can group numbers in a product in anyway you want and still get

the same answer. (Multiplication) Ex.:2x2(3y)=3y(2x2)

Closure Property – The sum or product of real numbers is also a real number.

Distributive Property (Distributive property of multiplication over addition) – If a term is multiplied

by terms in a parenthesis, we need to distribute the term over all the terms inside the parenthesis.

Identity Property – A number multiplied by 1 is the number itself. (Multiplication)

- A number added to zero is the number itself. (Addition)

Inverse Property – Any real number added/multiplied to its inverse is 0. Ex.: 3+(-3) = 0 (Addition)

Ex.: 3(1/3)=0

For Order of Operations: P.E.M.D.A.S. – Parenthesis, Exponent, Multiply, Divide, Add, Subtract.



By: J.G.M. Manuel

MODULE 2

Algebraic Expressions

Algebraic Expression – combination of constants and variables related by the operations.

Term - part of an algebraic expressions together with the plus or minus sign.

Variable – symbol which stand for any element in a given set. It can be letters such as x,y and z.

Constant - one possible value.

Example:

Algebraic Expression:

𝟑𝒙𝟐 + 𝟒𝒙 − 𝟔

Terms:

𝟑𝒙𝟐 , 4x , -6

Factor – a term consists of a product of two or more constants or variables.

Literal Factor – if the factor is a letter.

Numerical Factor – if the factor is a number.

Coefficient – the factor of a term of another factor.

Example:

Term: 7xyz

Factors: 7, x, y, and z

Numerical Factor: 7

Coefficient of yz: 7x

Coefficient of xy: 7z

Similar terms or Like terms – terms with same literal factors.

Monomial – one term

Example:

𝟐𝒙𝟐𝒚𝟑

Binomial – two terms

Example:

𝒙𝟐 − 𝟐𝟓

Trinomial – three terms

Example:

𝟓𝒙𝟏/𝟐 − 𝟐𝒙 − 𝟏

Multinomial – 2 or more terms

Example:

𝟑𝒙𝟐𝒚 − 𝟒𝒙𝟑 + 𝟒𝒚𝟐 − 𝟐



By: J.G.M. Manuel

Polynomial – algebraic expressions involve constant or variables with positive integral exponents

and no term with a variable in a denominator.

Examples:

Polynomials:

𝟐𝒙𝟐 − 𝟔𝒙 + 𝟖

𝟔𝒚𝟐 + 𝟒𝒚𝟑𝒛 + 𝟐

𝟏𝟒𝒙𝟐 + 𝟏𝟔𝒙 − 𝟐𝒙𝟑

Not Polynomials:

𝟏𝟎𝒙𝟑𝒚 + √𝒙 + 𝟑

𝟔𝒚𝟐 + 𝟐𝒚𝟐𝒛 + 𝟒𝒛−𝟐

𝟕𝒙𝟐𝒚𝟐 +𝟒

𝒚− 𝟐

Degree of the polynomial – highest degree of any of the terms in a polynomial. Degree is the

exponent of a variable.

Example:

𝟒𝒙𝟑𝒚𝟒𝒛 + 𝟐𝒙𝟐𝒚𝟐𝒛𝟐 − 𝒙𝒚

The degree with respect to x is 3.

The degree with respect to xy is 2.

The degree with respect to xyz is 8.

Evaluating Algebraic Expressions

-Replace the variables with the given value of variables, then perform the indicated

operation.

Example: Given: x=3, y=2, and z=-1

x3+y3+z3-2xyz

33+23+(-1)3-2(3)(2)(-1)

27+8-1+12

=46

Addition and Subtraction of Polynomials

combining like terms

1. Identify terms with same literal coefficient.

2. Find the sum of each group of like terms by adding their numerical coefficient.

Example:

Subtract 𝟐𝒙𝟐𝒚 + 𝟏𝟐 − 𝟓𝒙𝟑 − 𝟑𝒙𝒚 − 𝟐𝒙𝟐𝒚𝟐 from the sum of −𝟒𝒙𝟐𝒚 + 𝟖 + 𝟓𝒙𝒚 − 𝟐𝒙𝟑 −

𝟐𝒙𝟐𝒚𝟐 and 𝟕 − 𝟑𝒙𝟑 + 𝟐𝒙𝟐𝒚 + 𝒙𝟐𝒚𝟐 − 𝟖𝒙𝒚

= (−𝟒𝒙𝟐𝒚 + 𝟖 + 𝟓𝒙𝒚 − 𝟐𝒙𝟑 − 𝟐𝒙𝟐𝒚𝟐) + (𝟕 − 𝟑𝒙𝟑 + 𝟐𝒙𝟐𝒚 + 𝒙𝟐𝒚𝟐 − 𝟖𝒙𝒚) – (𝟐𝒙𝟐𝒚 + 𝟏𝟐 − 𝟓𝒙𝟑 − 𝟑𝒙𝒚 −

𝟐𝒙𝟐𝒚𝟐)

= (−𝒙𝟐𝒚𝟐 − 𝟐𝒙𝟐𝒚 − 𝟑𝒙𝒚 − 𝟓𝒙𝟑 + 𝟏𝟓) – (𝟐𝒙𝟐𝒚 + 𝟏𝟐 − 𝟓𝒙𝟑 − 𝟑𝒙𝒚 − 𝟐𝒙𝟐𝒚𝟐)

= 𝒙𝟐𝒚𝟐 − 𝟒𝒙𝟐𝒚 + 𝟑

Multiplication of Polynomials

1. Apply the distributive property, rules on exponent, and rules on signs. 2. Arrange the terms of polynomials in descending order.



By: J.G.M. Manuel

Division of Polynomials

-Arrange the terms descending order according to exponents.

-Divide each term of the polynomial by the monomial using the given law of exponent

-Simplify if possible.

Removal of symbols of grouping

Rules on removing symbols of grouping:

1. Removing a grouping symbol preceded by a (-) sign. Example:

𝟓𝒙– (𝟒𝒚 – 𝒛) = 𝟓𝒙 – 𝟒𝒚 + 𝒛

2. Removing a grouping symbol preceded by a (+) sign. Example:

𝟓𝒙 + (𝟒𝒚 – 𝒛) = 𝟓𝒙 + 𝟒𝒚 – 𝒛

3. Removing a grouping symbol preceded by a factor. Example:

−𝟐𝒙(𝟐𝒙 − 𝟑𝒙𝒚𝟐) = −𝟒𝒙𝟐 + 𝟔𝒙𝟐𝒚𝟐

𝟑𝒚(𝟓𝒙𝟐𝒚 − 𝟐𝒚𝟑 = 𝟏𝟓𝒙𝟐𝒚𝟑 − 𝟐𝒚𝟒

4. Removing grouping symbols within other grouping symbol.

Remove first the parenthesis ()

Then the braces []

The last is the {} Example:

𝒂{𝟐𝒂𝒃 – 𝟐[𝒂(𝟐𝒂𝒃 – 𝒃 + 𝒂𝒃) – 𝟐𝒂 (𝒂𝒃 + 𝟐) + 𝒂 ( 𝒃 – 𝟑𝒂)]}

𝒂{𝟐𝒂𝒃 – 𝟐[𝟐𝒂𝟐𝒃 − 𝒂𝒃 + 𝒂𝟐𝒃 − 𝟐𝒂𝟐𝒃 − 𝟒𝒂 + 𝒂𝒃 − 𝟑𝒂𝟐]} 𝒂{𝟐𝒂𝒃 − 𝟒𝒂𝟐𝒃 + 𝟐𝒂𝒃 − 𝟐𝒂𝟐𝒃 + 𝟒𝒂𝟐𝒃 − 𝟖𝒂 − 𝟐𝒂𝒃 + 𝟔𝒂𝟐} 𝟐𝒂𝟐𝒃 − 𝟒𝒂𝟑𝒃 + 𝟐𝒂𝟐𝒃 − 𝟐𝒂𝟑𝒃 + 𝟒𝒂𝟑𝒃 − 𝟖𝒂𝟐 − 𝟐𝒂𝟐𝒃 + 𝟔𝒂𝟑

𝟐𝒂𝟐𝒃 − 𝟐𝒂𝟑𝒃 − 𝟖𝒂𝟐 + 𝟔𝒂𝟑

SPECIAL PRODUCTS

Special Product is a multiplication of algebraic expression without using the long method.

1. Products of a Sum and Difference of Two Terms

Form: (𝒂 + 𝒃)(𝒂 − 𝒃) = 𝒂𝟐 − 𝒃𝟐 Example:

a) (𝟐𝒙 − 𝟓)(𝟐𝒙 + 𝟓) = 𝟒𝒙𝟐 − 𝟐𝟓

b) (𝟖𝒙𝟑 − 𝒚𝟒)(𝟖𝒙𝟑 + 𝒚𝟒) = 𝟔𝟒𝒙𝟔 − 𝒚𝟖

2. Square of a Binomial

Form: (𝒂 ± 𝒃)𝟐 = (𝒂𝟐 ± 𝒂𝒃 + 𝒃𝟐) Example:

a) (𝟑𝒙 − 𝟒)𝟐 = (𝟑𝒙)𝟐 − 𝟐(𝟑𝒙)(𝟒) + (𝟒)𝟐

= 𝟗𝒙𝟐 − 𝟐𝟒𝒙 + 𝟏𝟔

b) (𝟗𝒙𝟐𝒚𝟑 + 𝟒𝒙𝟓𝒚𝟕)𝟐

= (𝟗𝒙𝟐𝒚𝟑)𝟐 + 𝟐(𝟗𝒙𝟐𝒚𝟑)(𝟒𝒙𝟓𝒚𝟕) + (𝟒𝒙𝟓𝒚𝟕)𝟐

= 𝟖𝟏𝒙𝟐𝒚𝟔 + 𝟕𝟐𝒙𝟕𝒚𝟏𝟎 + 𝟏𝟔𝒙𝟏𝟎𝒚𝟏𝟒



By: J.G.M. Manuel

3. Cube of a Binomial

Form: (𝒂 ± 𝒃)𝟑 = 𝒂𝟑 ± 𝟑𝒂𝟐𝒃 + 𝟑𝒂𝒃𝟐 ± 𝒃𝟑 Example:

a) (𝟐𝒙 + 𝟓)𝟑

= (𝟐𝒙)𝟑 + 𝟑(𝟐𝒙)𝟐(𝟓) + 𝟑(𝟐𝒙)(𝟓)𝟐 + (𝟓)𝟑

= 𝟖𝒙𝟑 + 𝟔𝟎𝒙𝟐 + 𝟏𝟓𝟎𝒙 + 𝟏𝟐𝟓

4. Product of Two Dissimilar Binomials Form: (𝒂 + 𝒃)(𝒄 + 𝒅) = 𝒂𝒄 + 𝒂𝒅 + 𝒃𝒄 + 𝒃𝒅

FOIL method (first term-ac, outer term-ad, inner term-bc, last term-bd)

Example: a) (𝒙 − 𝟒)(𝒚 + 𝟐)

= 𝒙𝒚 + 𝟐𝒙 − 𝟒𝒚 − 𝟖

5. Square of Polynomials

Form: (𝒂 + 𝒃 + 𝒄)𝟐 = 𝒂𝟐 + 𝒃𝟐 + 𝒄𝟐 + 𝟐𝒂𝒃 + 𝟐𝒂𝒄 + 𝟐𝒃𝒄 Example:

a) (𝟐𝒙 − 𝟑𝒚 + 𝟒𝒛)𝟐

= 𝟒𝒙𝟐 + 𝟗𝒚𝟐 + 𝟏𝟔𝒛𝟐 − 𝟏𝟐𝒙𝒚 + 𝟏𝟔𝒙𝒛 − 𝟐𝟒𝒚𝒛

FACTORING

Factoring is the process of finding the factors.

1. Common Factor Form: 𝒂𝒃 − 𝒂𝒄 + 𝒂𝒅 = 𝒂(𝒃 − 𝒄 + 𝒅) Example:

a) 𝟑𝒚𝟐 + 𝟏𝟐𝒚

= 𝟑(𝒚𝟐 + 𝟒𝒚) →3 and 12 have a common factor of 3

= 𝟑𝒚(𝒚 + 𝟒) →y2 and 4y have also a common factor which is y

b) (𝒙 + 𝒚)𝟐 + 𝒛(𝒙 + 𝒚) = (𝒙 + 𝒚)(𝒙 + 𝒚 + 𝒛) →(x+y)2 and z(x+y) have a common factor of (x+y)

2. Difference of two squares

Form: 𝒂𝟐 − 𝒃𝟐 = (𝒂 + 𝒃)(𝒂 − 𝒃) Example:

a) 𝒙𝟐 − 𝟏𝟔

= (𝒙 + 𝟒)(𝒙 − 𝟒) →The √𝒙𝟐 is x and √𝟏𝟔 is 4.

b) 𝟒𝒙𝟐 − 𝟗 = (𝟐𝒙 + 𝟑)(𝟐𝒙 − 𝟑) →get the square root of 4x2 and 9

3. Sum or difference of two cubes

Form: 𝒂𝟑 ± 𝒃𝟑 = (𝒂 ± 𝒃)(𝒂𝟐 ∓ 𝒂𝒃 + 𝒃𝟐) Example:

a) 𝒙𝟑 − 𝟖

= (𝒙)𝟑 − (𝟐)𝟑

= (𝒙 − 𝟐)(𝒙𝟐 + 𝟐𝒙 + 𝟐𝟐)



By: J.G.M. Manuel

= (𝒙 − 𝟐)(𝒙𝟐 + 𝟐𝒙 + 𝟒)

b) 𝒙𝟑𝒚𝟔 + 𝟔𝟒

= (𝒙𝒚𝟐)𝟑 − (𝟒)𝟑

= (𝒙𝒚𝟐 + 𝟒)((𝒙𝒚𝟐)𝟐 − (𝒙𝒚𝟐)(𝟒) + 𝟒𝟐)

= (𝒙𝒚𝟐 + 𝟒)(𝒙𝟐𝒚𝟒 − 𝟒𝒙𝒚𝟐 + 𝟏𝟔)

4. Sum or difference of odd powers

Form: (𝒂𝒏 ± 𝒃𝒏) = (𝒂 ± 𝒃)(𝒂𝒏−𝟏 ∓ 𝒂𝒏−𝟐𝒃 + 𝒂𝒏−𝟑𝒃𝟐 ∓ ⋯ ± 𝒃𝒏−𝟏) Example:

a) 𝒙𝟓 + 𝟑𝟐

= 𝒙𝟓 + (𝟐)𝟓

= (𝒙 + 𝟐)(𝒙𝟒 − 𝟐𝒙𝟑 + (𝟐)𝟐𝒙𝟐 − (𝟐)𝟑𝒙 + (𝟐)𝟒)

= (𝒙 + 𝟐)(𝒙𝟒 − 𝟐𝒙𝟑 + 𝟒𝒙𝟐 − 𝟖𝒙 + 𝟏𝟔)

b) 𝒙𝟓 − 𝟑𝟐

= 𝒙𝟓 − (𝟐)𝟓

= (𝒙 − 𝟐)(𝒙𝟒 + 𝟐𝒙𝟑 + (𝟐)𝟐𝒙𝟐 + (𝟐)𝟑𝒙 + (𝟐)𝟒)

= (𝒙 − 𝟐)(𝒙𝟒 + 𝟐𝒙𝟑 + 𝟒𝒙𝟐 + 𝟖𝒙 + 𝟏𝟔)

5. Perfect square trinomial

Form: 𝒂𝟐 ± 𝟐𝒂𝒃 + 𝒃𝟐 = (𝒂 ± 𝒃)𝟐 Example:

a) 𝒙𝟐 − 𝟏𝟐𝒙 + 𝟑𝟔 →the first term, x2, is the square of x and the last term, 36, is

= (𝒙 − 𝟔)𝟐 the square of 6. Since the middle term has a “minus sign”, the 36 is the square of -6.

b) 𝟗𝒙𝟐 + 𝟐𝟒𝒙 + 𝟏𝟔

= (𝟑𝒙 + 𝟒)𝟐

6. Quadratic trinomial

Form: 𝒂𝒄𝒙𝟐 + (𝒂𝒅 + 𝒃𝒄)𝒙 + 𝒃𝒅 = (𝒂𝒙 + 𝒃)(𝒄𝒙 + 𝒅) Example:

a) 𝟓𝒙𝟐 + 𝟏𝟏𝒙 + 𝟐 →find the product of the first and last term: (5)(2)=10

= 𝟓𝒙𝟐 + 𝟏𝒙 + 𝟏𝟎𝒙 + 𝟐 →Think of two factors of 10 that adds up to 11: 1 and 10

= (𝟓𝒙𝟐 + 𝟏𝒙) + (𝟏𝟎𝒙 + 𝟐) →Group the two pairs of terms then remove common factors = 𝒙(𝟓𝒙 + 𝟏) + 𝟐(𝟓𝒙 + 𝟏) →then we can factor out the common factor (5x+1)

= (𝟓𝒙 + 𝟏)(𝒙 + 𝟐)

b) 𝟒𝒙𝟐 + 𝟕𝒙 − 𝟏𝟓 →find the product of the first and last term: (4)(15)= - 60

= 𝟒𝒙𝟐 − 𝟓𝒙 + 𝟏𝟐𝒙 − 𝟏𝟓 →Think of two factors of - 60 that adds up to 7: -5 and 12

= (𝟒𝒙𝟐 − 𝟓𝒙) + (𝟏𝟐𝒙 − 𝟏𝟓) →Group the two pairs of terms then remove common factors = 𝒙(𝟒𝒙 − 𝟓) + 𝟑(𝟒𝒙 − 𝟓) →then we can factor out the common factor (4x-5) = (𝟒𝒙 − 𝟓)(𝒙 + 𝟑)

7. Factoring by adding or subtracting perfect square Example:

Dapat ganito ang perfect square

trinomial, kaya ang kulang niya ay 4x2



By: J.G.M. Manuel

a) 𝒙𝟒 + 𝟒 ↓ ↓

𝒙𝟐 𝟐𝒙𝟒 + 𝟒𝒙𝟐 + 𝟒

= (𝒙𝟒 + 𝟒𝒙𝟐 + 𝟒) − 𝟒𝒙𝟐 →add and subtract 4x2

= (𝒙𝟐 + 𝟐)𝟐 − (𝟐𝒙)𝟐 →then, factor it using the difference of two square

= (𝒙𝟐 + 𝟐 + 𝟐𝒙)(𝒙𝟐 + 𝟐 − 𝟐𝒙)

8. Factoring by grouping

Example:

a) 𝒙𝟑 + 𝟐𝒙𝟐 + 𝟖𝒙 + 𝟏𝟔

= 𝒙𝟐(𝒙 + 𝟐) + 𝟖(𝒙 + 𝟐)

= (𝒙 + 𝟐)(𝒙𝟐 + 𝟖)

Rational Expression

-is a fraction whose numerator and denominator are polynomials -can be simplified by applying factoring Equivalent Fraction – the numerator and denominator of a fraction may be multiplied or divided by a nonzero number without changing the value of the fraction.

𝒂

𝒃=

𝒂𝒄

𝒃𝒄

𝒂

𝒃=

𝒂𝒄𝒃𝒄

Example:

𝟑𝒙

𝟑(𝒙+𝟑)=

𝒙

𝒙+𝟑

Simplifying Fractions 1. Factor all expression in numerator and denominator 2. Apply multiplicative cancellation Example:

(𝒙𝟐−𝟑𝟔)

𝒙−𝟔=

(𝒙+𝟔)(𝒙−𝟔)

𝒙−𝟔

= 𝒙 + 𝟔 Multiplication and Division of Rational Expression 1. Factor all factorable algebraic expression on the given 2. Arrange the factors in the correct position, factor other factorable algebraic expression if possible. 3. Multiplicative cancellation, after cancellation combine all the expressions left.

𝒂

𝒃∙

𝒄

𝒅=

𝒂𝒄

𝒃𝒅

Example:

(𝒙𝟐−𝟐𝟓)

𝟒∙

𝟐𝒙𝟐

(𝑿−𝟓)=

(𝑿+𝟓)(𝑿−𝟓)

𝟒∙

𝟐𝒙𝟐

𝑿−𝟓

=𝒙𝟐(𝑿+𝟓)

𝟐

There are no common factors to all four terms. So,

we try grouping the first two together and the last

two together, and factor out the common factor.



By: J.G.M. Manuel

=𝒙𝟑+𝟓𝒙𝟐

𝟐

Least Common Multiple (LCM) – the product of all the prime factors, with each raised to the highest power that occurs in any expression. Addition and Subtraction of Rational Expression 1. Factor completely all the denominators 2. Find the LCM for all denominators 3. Combine all the fractions to one single fraction 4. Distribute the numbers so that there would be no parenthesis, then combine like terms 5. Factor the remaining and apply multiplicative cancellation. Example: Complex Fraction - both numerator and denominator are fraction. 1. Search for the longest line, above it was the numerator and below it is the denominator. 2. Applying the rules in addition, subtraction, multiplication and division of rational expression, simplify the numerator and denominator to its simplest form. 3. Multiply the simplified form of numerator with the reciprocal of the denominator.

RADICALS Radical is a number that has a fractional exponent

√𝒙𝒏

Laws of Radicals

1. √𝒙𝒏

∙ √𝒚𝒏 = √𝒙𝒚𝒏

Example:

a) √𝟑 ∙ √𝟓 = √𝟏𝟓

b) √𝟏𝟒𝟑

∙ √𝟐𝟑

= √𝟐𝟖𝟑

2. √𝒙

𝒏

√𝒚𝒏 = √𝒙

𝒚

𝒏

Example:

a) √𝟐𝟎

𝟓

√𝟒𝟓 = √

𝟐𝟎

𝟒

𝟓= √𝟓

𝟓

b) √𝒙𝒚𝒛

√𝒂𝒃= √

𝒙𝒚𝒛

𝒂𝒃

3. √𝒙𝒎𝒏= ( √𝒙

𝒏)

𝒎

a) √𝒂𝟑𝟒= ( √𝒂

𝟒)𝟑

4. √ √𝒙𝒏𝒎

= √𝒙𝒎∙𝒏

a) √ √𝒂𝒃𝟒𝟓

= √𝒂𝒃𝟐𝟎

Simplifying Radicals

1. Radicand has no factor raised to a power greater than or equal to the index of radical. Example:

a) √𝟐𝟓𝟔𝒙𝟕𝒚𝟒𝒛𝟔𝟑

= √𝟒𝟒𝒙𝟕𝒚𝟒𝒛𝟔𝟑

= 𝟒𝒙𝟐𝒚𝒛𝟐 √𝟒𝒙𝒚𝟑

Index

Radicand

Radical sign



By: J.G.M. Manuel

2. No fraction appears on radicand.

Example:

a) √𝒂

𝒃𝒄

= √𝒂

𝒃𝒄∙

𝒃𝒄

𝒃𝒄

=√𝒂𝒃𝒄

𝒃𝒄

3. No radicals within a radical Operations Involving Radicals Addition and Subtraction of Radicals → Terms with same index and radicands may be combined Example:

a) √𝟑 − √𝟐𝟕 + √𝟒𝟖

= √𝟑 − √𝟑𝟑 + √𝟒𝟐 ∙ 𝟑

= √𝟑 − 𝟑√𝟑 + 𝟒√𝟑

= 𝟐√𝟑

Multiplication of Radicals 1. Same Indices

Ex. √𝟒 ∙ √𝟕 = √𝟒 ∙ 𝟕

= √𝟐𝟖 2. Different Indices, but same Radicands

Ex. √𝟒 ∙ √𝟒𝟑

= √𝟒𝟐+𝟑𝟐∙𝟑

= √𝟒𝟓𝟔

= √𝟏𝟎𝟐𝟒𝟔

3. Different Indices and radicands

Ex. √𝟑 ∙ √𝟓𝟑

= √𝟑𝟑 ∙ 𝟓𝟐𝟐∙𝟑

= √𝟐𝟕 ∙ 𝟐𝟓𝟔

= √𝟔𝟕𝟓𝟔

Division of Radicands

- rewrite the expressions to fractional form and then use rationalization

Ex. √𝟑𝟐𝒂𝟗

𝟐𝟕𝒃𝟐

𝟒= √𝟐𝟒𝒂𝟖𝟐𝒂

𝟑𝟑𝒃𝟐

𝟒

= 𝟐𝒂𝟐 √𝟐𝒂∙𝟐𝒂

𝟑𝟑𝒃𝟐 ∙𝟑𝒃𝟐

𝟑𝒃𝟐

𝟒

= 𝟐𝒂𝟐 √𝟔𝒂𝒃𝟐

𝟑𝟒𝒃𝟒

𝟒

=𝟐𝒂𝟐

𝟑𝒃√𝟔𝒂𝒃𝟐𝟒

College Algebra

Education

Transcript of College Algebra