Your Algebra Homework Can Now Be Easier Than Ever!

Equivalent Forms of Algebraic Expressions

Examples

A. Long Division

B. Multiplying by "1"

C. Completing the Square

Practice Problems

Algebra can be used to change the form of many fractional expressions .

Examples:

Basic simplification

"Rationalizing"
Mult. by 1

Factoring highest power
from numerator and denominator


"Do the examples above bring back old (fond?) memories of LIMITS?"

  Separation

Long Division

Mult. by 1

Mult. by 1

Before we comment on the algebra, let's examine these examples in a cal-
culus context. Since our virtual calculus site is still in development, you'll
need paper and pencil.

Exercise 1: Evaluate the following.

Answers Answers

Note that algebra is essential in evaluating the above limits and integrals.
Another way to evaluate would be as follows:

Just for fun (we're not afraid to use fun and calculus in the same sentence),
show both answers to Exercise 1(g) are equivalent .

A. Long Division

A fraction ( rational expression ) that has a monomial denominator, like
Example 4, easily separates. However, if the fraction is improper (degree of
the numerator is greater than or equal to degree of the denominator), then
dividing by a polynomial requires Long Division. Here's Example 5 worked
out.

B. Multiplying by "1"

Of our original examples, three of them (#2, 6, and 7) fall into this category.
The integral for sec x is derived using multiplication by a form of 1.

C. Completing the Square

In your previous math courses, completing the square was used to solve
quadratics or to change the form of a second degree polynomial of the type
Ax2 + By2 + Cx + Dy + E = 0. In Calc II, completing the square will be a
useful (and necessary) tool in evaluating certain integrals.

Illustrations: Complete the square.

Are you following the process? Maybe you need to try this yourself.

Exercise 2: Rewrite each expression by completing the square.

Answers

Now let's try to complete the square starting with the quadratic form
y = ax2 + bx + c.

We have completed the square, but what good is this expression? Set y = 0
and find x- intercepts :

Now you know why math instructors have such a high regard for completing
the square.

We will finish this section with examples of two integrals .

Example.

let

Example.

Unlike our first example, substitution won 't work. Let's complete the square
and see what follows.

Recall 

Conclusion: This review topic (as well as Topics 3 and 4) demonstrates
how algebra is used to evaluate limits and integrals. The challenge is not
just knowing the algebra, but being able to recognize when to apply it. That
only comes from practice.

Practice Problems.

2.1 Find the quotient and remainder:



Answers

2.2 All of the following problems require multiplication by a form of 1

Answers
Answers
Answers
Answers

2.3 Complete each square.

Answers

Answers

Answers

Answers to Practice Problems.

Return to Problem

Return to Problem

Return to Problem

Return to Problem

Return to Problem

Return to Problem

Return to Problem

Return to Problem

Return to Problem

Beginning of Topic

250 Review Topics

250 Skills Assessment

Answers

Recall that for r > 0,
as n→∞
OR
"When denominator grows
fraction approaches 0"

Alternate Method :


Return to Review Topic

Answers:

e)

f) Let u = ex + 1. Why is
Ans:

g) Let u = 1+sinx.
Ans:

Return to Review Topic

Answers:

Return to Review Topic

Prev Next

Start solving your Algebra Problems in next 5 minutes!

Algebra Helper
Download (and optional CD)

Only $39.99

Click to Buy Now:


OR

2Checkout.com is an authorized reseller
of goods provided by Sofmath

Attention: We are currently running a special promotional offer for Algebra-Answer.com visitors -- if you order Algebra Helper by midnight of November 2nd you will pay only $39.99 instead of our regular price of $74.99 -- this is $35 in savings ! In order to take advantage of this offer, you need to order by clicking on one of the buttons on the left, not through our regular order page.

If you order now you will also receive 30 minute live session from tutor.com for a 1$!

You Will Learn Algebra Better - Guaranteed!

Just take a look how incredibly simple Algebra Helper is:

Step 1 : Enter your homework problem in an easy WYSIWYG (What you see is what you get) algebra editor:

Step 2 : Let Algebra Helper solve it:

Step 3 : Ask for an explanation for the steps you don't understand:



Algebra Helper can solve problems in all the following areas:

  • simplification of algebraic expressions (operations with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots (radicals), absolute values)
  • factoring and expanding expressions
  • finding LCM and GCF
  • (simplifying, rationalizing complex denominators...)
  • solving linear, quadratic and many other equations and inequalities (including basic logarithmic and exponential equations)
  • solving a system of two and three linear equations (including Cramer's rule)
  • graphing curves (lines, parabolas, hyperbolas, circles, ellipses, equation and inequality solutions)
  • graphing general functions
  • operations with functions (composition, inverse, range, domain...)
  • simplifying logarithms
  • basic geometry and trigonometry (similarity, calculating trig functions, right triangle...)
  • arithmetic and other pre-algebra topics (ratios, proportions, measurements...)

ORDER NOW!

Algebra Helper
Download (and optional CD)

Only $39.99

Click to Buy Now:


OR

2Checkout.com is an authorized reseller
of goods provided by Sofmath
Check out our demo!
 
"It really helped me with my homework.  I was stuck on some problems and your software walked me step by step through the process..."
C. Sievert, KY
 
 
Sofmath
19179 Blanco #105-234
San Antonio, TX 78258
Phone: (512) 788-5675
Fax: (512) 519-1805
 

Home   : :   Features   : :   Demo   : :   FAQ   : :   Order

Copyright © 2004-2024, Algebra-Answer.Com.  All rights reserved.