There, we have seen some special products as Algebra Formulas and we have also seen their proofs.

Now we make use of some of those Formulas to find the factors, given the product.

We list out the required Formulas from there with L.H.S. and R.H.S. reversed.

We have to remember these Formulas from L.H.S. to R.H.S. and from R.H.S. to L.H.S. and be able to apply them in both directions.

Algebra Formulas used for Factoring Binomials

Formula 1 in Algebra Factoring:

Difference of Two Squares as Product of Sum and Difference:

a^{2} - b^{2} = (a + b)(a - b)

Formula 2 in Algebra Factoring:

Sum of Two Cubes as Product of Two Factors:

a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})

Formula 3 in Algebra Factoring:

Difference of Two Cubes as Product of Two Factors:

a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})

Here a, b, c, d, x are all real numbers.

Each of the letters in fact represent a TERM.

e.g. The above Formula 1 can be stated as

(First term)^{2} - (Second term)^{2} = (First term + Second term)(First term - Second term)

Similarly in other Formulae also, we can replace each of the letters by a TERM.

Get The Best Grades With the Least Amount of Effort : Factoring Binomials

Here is a collection of proven tips, tools and techniques to turn you into a super-achiever - even if you've never thought of yourself as a "gifted" student.

The secrets will help you absorb, digest and remember large chunks of information quickly and easily so you get the best grades with the least amount of effort.

If you apply what you read from the above collection, you can achieve best grades without giving up your fun, such as TV, surfing the net, playing video games or going out with friends!

Let P = 7a^{4} + 28 = 7(a^{4} + 4) = 7{(a^{2})^{2} + 2^{2}} Here, inside the bracket, (first term)^{2} + (second term)^{2} is present. Let us add and subtract 2(first term)(second term), so that the value does n't change. i.e. Adding and subtracting 2(a^{2})(2), we get P = 7{(a^{2})^{2} + 2^{2} + 2(a^{2})(2) - 2(a^{2})(2)} = 7{(a^{2} + 2)^{2} - 2(a^{2})(2)} = 7 {(a^{2} + 2)^{2} - (2a)^{2}} This is the difference of the squares of two terms which is equal to theproduct of the sum and difference of the terms.[See Formula 3] ∴ P = 7{(a^{2} + 2) + (2a)}{(a^{2} + 2) - (2a)} P = 7(a^{2} + 2a + 2)(a^{2} - 2a + 2).

Thus Algebra Factoring of 7a^{4} + 28 by using Algebra Formulas gave the factors as 7(a^{2} + 2a + 2)(a^{2} - 2a + 2). Ans.

Let P = x^{12} - 4096 We know 4096 = 8 x 512 = 8 x 8 x 64 = 8^{4} = (2^{3})^{4} = 2^{(3 x 4)} = 2^{12} ∴ P = x^{12} - 4096 = x^{12} - 2^{12} = (x^{6})^{2} - (2^{6})^{2} This is difference of squares of two terms which is equal to the product of the sum and difference of the two terms [See Formula 3]. ∴ P = (x^{6} + 2^{6})(x^{6} - 2^{6})= {(x^{2})^{3} + (2^{2})^{3}}{(x^{3})^{2} - (2^{3})^{2}} The first bracket is sum of two cubes to which we can apply Formula 4 and the second bracket is difference of two squares to which we can apply Formula 3. We have a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2}) [See Formula 4] a^{2} - b^{2} = (a + b)(a - b) [See Formula 3] Applying these here, we get P = {(x^{2}) + 2^{2})}{(x^{2})^{2} - (x^{2})(2^{2}) + (2^{2})^{2}}{(x^{3}) + (2^{3})}{(x^{3}) - (2^{3})} The last two brackets are sum of two cubes (Formula 4) and difference of two cubes (Formula 5) a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2}) [See Formula 5] Applying Formula 4 and 5 to last two brakets, we get {(x^{3}) + (2^{3})} = (x + 2)(x^{2} - x(2) + 2^{2}) = (x + 2)(x^{2} - 2x + 4) {(x^{3}) - (2^{3})} = (x - 2)(x^{2} + x(2) + 2^{2}) = (x - 2)(x^{2} + 2x + 4) ∴ P = (x^{2} + 4)(x^{4} - 4x^{2} + 16)(x + 2)(x^{2} - 2x + 4)(x - 2)(x^{2} + 2x + 4).

Thus Algebra Factoring of x^{12} - 4096by using Algebra Formulas gave the factors as (x^{2} + 4)(x^{4} - 4x^{2} + 16)(x + 2)(x^{2} - 2x + 4)(x - 2)(x^{2} + 2x + 4). Ans.

Exercise : Factoring Binomials

Solve the following problems on Factoring Binomials.

Factorize x^{4} + 4

Factorize 729a^{6} - 64b^{6}

For Answers See at the bottom of the Page.

Progressive Learning of Math : Factoring Binomials

Recently, I have found a series of math curricula (Both Hard Copy and Digital Copy) developed by a Lady Teacher who taught everyone from Pre-K students to doctoral students and who is a Ph.D. in Mathematics Education.

This series is very different and advantageous over many of the traditional books available. These give students tools that other books do not. Other books just give practice. These teach students “tricks” and new ways to think.

These build a student’s new knowledge of concepts from their existing knowledge. These provide many pages of practice that gradually increases in difficulty and provide constant review.

These also provide teachers and parents with lessons on how to work with the child on the concepts.

The series is low to reasonably priced and include

Are you spending lot of money for math tutors to your child and still not satisfied with his/her grades ?

Do you feel that more time from the tutor and more personalized Math Help to identify and fix the problems faced by your child will help ?

Here is a fool proof solution I strongly recommend and that too With a minuscule fraction of the amount you spent on tutors with unconditional 100% money back Guarantee, if you are not satisfied.

It is like having an unlimited time from an excellent Tutor.

It is an Internet-based math tutoring software program that identifies exactly where your child needs help and then creates a personal instruction plan tailored to your child’s specific needs.

If your child can use a computer and access the Internet, he or she can use the program. And your child can access the program anytime from any computer with Internet access.

There is an exclusive, Parent Information Page provides YOU with detailed reports of your child’s progress so you can monitor your child’s success and give them encouragement. These Reports include

Time spent using the program

Assessment results

Personalized remediation curriculum designed for your child

Details the areas of weakness where your child needs additional help

Provides the REASONS WHY your child missed a concept

List of modules accessed and amount of time spent in each module

Quiz results

Creates reports that can be printed and used to discuss issues with your child’s teachers

These reports are created and stored in a secure section of the program, available exclusively to you, the parent. The section is accessed by a password that YOU create and use. No unauthorized users can access this information.

Its research-based results have proven that it really works for all students! in improving math skills and a TWO LETTER GRADE INCREASE in math test scores!,if they invest time in using the program.

Proven for More than 10,000 U.S. public school students who increased their math scores.