Solve by Factoring in Elementary Algebra

Learning how to solve equations is one of our main goals in Algebra. At this point, we know how to solve linear equations,

And now we begin to learn how to solve quadratic equations,

One way to solve quadratic equations is to use factoring and the zero-product property. The zero-product property sounds fancy but it simply states that,

So if we can factor a quadratic equation in standard form, equal to zero, we can then set each factor equal to zero and solve the resulting linear equations. For example, solve the following quadratic equation:

Step 1: Express the quadratic equation in standard form, equal to zero.

Step 2: Factor the quadratic expression.

Step 3: Apply the zero-product property and set each variable factor equal to zero.

Step 4: Solve the resulting linear equations.

The solutions are -5, and -3/2. You can check that these are solutions by substituting them into the original equation and verify that they lead to a true statement.  Here is another example on video:

Hopefully you can see that the factoring step is very important in this process. Take care to factor your quadratic equations carefully because this is the step where most students make an error.

• Elementary Algebra Textbook - Chapter 6 Factoring and Solving by Factoring
• Elementary Algebra Textbook - Chapter 6 YouTube Playlist

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Factoring in Elementary Algebra

Right now we are trying to learn how to factor in our Elementary Algebra course (Algebra 1).  Many students find this topic to be very difficult... but it is important to get it right because factoring is an essential skill in mathematics. For example,

The general factoring guidelines found in the free Elementary Algebra textbook offered on the Flat World Knowledge website follow:

1. Check for common factors. If the terms have common factors, then factor out the greatest common factor (GCF) and look for the resulting polynomial factors to factor further.
2. Determine the number of terms in the polynomial:

• Four term polynomials are factored by grouping.
• Trinomials (3 terms) are factored using “trial and error” or the AC method.
• Binomials (2 terms) are factored using the following special products.

3. Look for factors that can be factored further.
4. Check by multiplying.


* If a binomial is both difference of squares and cubes, then we should factor it as difference of squares first to obtain a more complete factorization.
* Not all polynomials with integer coefficients factor, in this case, we say that the polynomial is prime.

Solution: This four term polynomial has a GCF of 3x , we factor this out first.

We now factor the resulting four term polynomial by grouping.

The factor x^2 - 1 is a difference of squares and can be factored further.

And therefore we have,

You can check by multiplying.  A similar problem can be found on YouTube:

You will find many more embedded videos in the Elementary Algebra textbook, here are links to the appropriate sections:

6.1 Introduction to Factoring
6.2 Factoring Trinomials x^2 + bx + c
6.3 Factoring Trinomials ax^2 + bx + c
6.4 Factoring Binomials
6.5 General Guidelines for Factoring Polynomials

Learning how to factor takes lots of time and practice.

Quadratic Expression Generating Widget

Check out Wolfram Alpha's gallery of widgets. You can embed them into your blog or website as I have below.

Also, Wolfram Alpha Widget Builder allows you to easily generate widgets.  I was able to create my first widget in less than 20 minutes using their easy step oriented widget creating interface.  Enter the roots and click multiply... try as many real and complex roots as you wish!

Related articles

• Wolfram Alpha for Your Desktop (freetech4teachers.com)
• Introducing Wolfram|Alpha Widgets for Everyone, Everywhere
• A Computational Knowledge Engine on Your Site: Wolfram Alpha Launches Widget Builder

Low Tech Methods to Improve Success

I am always looking for ways to enhance the student experience with technology. I find that tech adds interest and can engage the student in ways that I could not in the past. In addition, the ability to understand and implement new technological tools is an important skill to develop in order to be competetive in the 21st century.

HOWEVER, it is also important to be reminded of the basics that we may take for granted. Co-author of College Success, Tom Lochhaas, does just that in his blog:

10 Tips for Getting the Most Out of Going (or Returning) to College

An excellent common sense reminder students should read and reread every semester!

Stay or Switch?

A very well done animated explanation of the classic Monty Hall problem.  Play the game and get multiple explanations on reasons why you should always switch!

http://www.stayorswitch.com/

Isaac Newton's Principia

Isaac Newton's Principia is recognized as the greatest scientific book ever written.  You can read it on Google books for free:  Newton's Principia with notes and illustrations.

Image via Wikipedia

In addition,  you can read about Isaac Newton here and view his childhood home below.

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FWK Technology

An excellent interview with Jon Williams the Chief Technology Officer of flat world knowledge.

Interview: Ed Tech Leader, Jon Williams of Flat World Knowledge

The "make it your own" MIYO platform has great potential. As Jon puts it, "one benefit of thinking of systems as a “platform” is that it forces you to take a longer view, and plan for future business growth and needs."

MIYO

Related articles

• Flat World Knowledge launches platform for customizable digital textbooks (teleread.com)
• Interview with Eric Frank, Flat World Knowledge, by Sue Polanka (teleread.com)
• Random House Invests In Flat World Knowledge (paidcontent.org)