college algebra

Module 6: Discussion on Squaring Binomials

 Module 5:  Mathematical Thinking about Binomials

Discussion Instructions:

Please review  the class discussion directions . They tell you how to prepare and reflect before responding to the prompt below. They also include detailed guidelines how you must respond to classmates and then reply to the responses you receive. Please pay close attention to the response directions.  To earn your discussion points, you must post, respond, and reply as directed.

Discussion writing prompts

Please create a new forum thread for each question:

· When you square a binomial like this  (1 + √x), what is a common mistake to make? Explain and detail your answer.

Response focus

1. In your response, in addition to the standard requirement of noting what you have in common and where you differ from classmates, offer a suggestion on how to improve your classmates' comfort level with the topics we are studying this week.

Remember to review the  academic expectations  for your submission.

Submission reminders:

· Submit your initial discussion post by 11:59 pm ET on Wednesday.  Respond to two of your classmates' discussion posts, and reply to the responses you received by 11:59 pm ET on Sunday.

· Contribute a minimum of 250 words to the initial post.

· Follow established  netiquette (Links to an external site.)  guidelines when participating in forums.

 

Reply

 Module 6:  Radical Expressions, Equations, and Functions

 

Root and Radicals

 

 

                                                                                         

 

 

Square Roots: Is a radical expression that has an index of 2. Generally, the index of a square root is not written: x=x2

 

For any positive real number  a:

· The principal or positive square root of  a, written  a , is the positive number  b such b2 = a

Example: 36 = 6

Cubic Root: The cube root of a number  a, written a3, is the number  b such b3 = a

Example: 83= 2      since 23 = 8

 

Root Functions Example: 

For this function f(x) = 11x−2 find f(6)

f(6) = 11(6)−2=64=8

Rational Exponents

 

Exponential form of an=a1/n

· When  a is nonnegative,  n can be any index

· When  a is negative,  n must be odd

 

Examples:

  a)7=71/2 

  b)15xy3=15xy1/3 

Simplifying Radicals:

A  perfect power is a number or expression that can be written as an expression raised to the power that is a whole number greater than 1.

A  perfect square is a number or expression that can be written as a square of an expression. A perfect square is a perfect second power.

A  perfect cube is a number or expression that can be written as a cube of an expression. A perfect cube is a perfect third power.

 

Examples:

1) Perfect squares: 

 

1=(12),4=(22),9=(32), ……….

x2=(x)2,x4=(x2)2,x6=(x3)2……….

 

2) Perfect cubes:

1=(13),8=(2)3,27=(3)3, ……….

x3=(x)3,x6=(x2)3,x9=(x3)3……….

 

Simplify Radicals using the product rule for radicals

 

Product rule: for nonnegative real numbers  a and  b

abn=anbn

 

Example

203=13203

x73=x23x53

322=16.2=16.2=42

Simplify Radicals using the quotient rule for radicals

Quotient rule: for nonnegative real numbers  a and  b

abn=anbn,b≠0

Examples:

1) 925=925=35

2) 24x3x=24x3x3=83=2

Adding, Subtracting, and Multiplying Radicals:

Add and Subtract Radicals:

· Simplify each radical expression

· Combine like radicals (if there are any)

Examples:

1) Simplify 3+27=3+3.9=3+3.3=43

2) Simplify x2−x2y+xy=x−x2.y+xy=X−xy+xy=X

 

Multiply Radicals

 

To multiply radicals, we use the product rule, and we can often simplify the new radical.

Examples: 

1)2x(8x−50)

Begin by using the distributive property: 

2x∗8x−2x∗50=16x2−100x=4x−100x=4x−10x

 

2) 6x38x6

Begin by using the product rule for radicals:

6x38x6=48x9=3x16x8=3x16x8=4x43x

Dividing Radicals

Rationalize Denominators:

Multiply both the numerator and denominator of the fraction by a radical that will produce a radicand in the denominator that is a perfect power for the index.

 

Examples:

1. Simplify:15 

15=15∗55=525=55

2) Simplify: 16a43b3

There are no common factors in the numerator and denominator. Let’s simplify the numerator before we rationalize the denominator.

 

16a43b3=8a32a3b3=2a2a3b3

Since the denominator contains  b and b3 is required, we need to more factors of  b or b2. Therefore, we multiply both numerator and denominator by b23

2a2a3b3=2a2a3b3b23b23=2a2ab23b33=2a2ab23b

Solving Radical Equations

A  radical equation is an equation that contains a variable in a radicand.

Steps:

1. Rewrite the equation so that one radical containing a variable is by itself (isolated) on one side of the equation.

2. Raise each side of the equation to a power equal to the index of the radical.

3. Combine like terms.

4. If the equation still contains a term with a variable in a radicand, repeat steps 1 through 3.

5. Solve the resulting equation for the variable.

6. Check all solutions in the original equation for extraneous solutions.

 

Examples: 

1. x=5⟶(x)2=52

X = 25

2. x−4−6=0

x−4=6

=(x−4)2=62

X – 4 = 36

X = 40

3. For f(x) = 5x−1−3x−2, find all values of x for which f(x) = 1

 

1 = 5x−1−3x−2

Isolate 5x−1

Then, 5x−1=1+3x−2

=  (5x−1)2=(1+3x−2)2

5x−1=(1+3x−2)(1+3x−2)

5x−1=1+3x−2+3x−2+(3x−2)2

5x−1=1+23x−2+3x−2

2x=23x−2

x=3x−2

Square both sides: x2=(3x−2)2⟶x2=3x−2

x2−3x+2=0

(x – 2) (x – 1) = 0

x = 2  or x = 1

References

Angel, A., & Rundle, D. (2018).  Intermediate algebra for college students (10th edition). Pearson.