Can u help me with definition of special products?

Polynomials that have to be squared, cubed etc. and solved mentally

Posted on Feb 09, 2011

Special product is RTOOOOT!! :)

Posted on Aug 05, 2012

SOURCE: definition of special product in

**Product** means the result you get after multiplying.

In Algebra **xy** means **x** multiplied by **y**

Likewise when you see **(a+b)(a-b)** it means** (a+b)**** **multiplied by** (a-b)**, which we will be using a lot here!

Special Binomial Products
So when you multiply binomials you get ... Binomial Products

And we are going to look at **three special cases** of multiplying binomials ... so they are **Special Binomial Products**.

1. Multiplying a Binomial by Itself
What happens when you square a binomial (in other words, multiply it by itself) .. ?

(a+b)2 = (a+b)(a+b) = ... ?

The result:

(a+b)2 = a2 + 2ab + b2

You can easily see why it works, in this diagram:

2. Subtract Times Subtract
And what happens if you square a binomial with a **minus** inside?

(a-b)2 = (a-b)(a-b) = ... ?

The result:

(a-b)2 = a2 - 2ab + b2

3. Add Times Subtract
And then there is one more special case... what if you multiply (a+b) by (a-b) ?

(a+b)(a-b) = ... ?

The result:

(a+b)(a-b) = a2 - b2

That was interesting! It ended up very simple.

And it is called the "**difference of two squares**" (the two squares are **a2** and **b2**).

This illustration may help you see why it works:

a2 - b2 is equal to (a+b)(a-b)
Note: it does not matter if (a-b) comes first:

(a-b)(a+b) = a2 - b2

The Three Cases
Here are the three results we just got:

(a+b)2
= a2 + 2ab + b2
} (the "perfect square trinomials")
(a-b)2
= a2 - 2ab + b2
(a+b)(a-b)
= a2 - b2
(the "difference of squares")
Remember those patterns, they will save you time and help you solve many algebra puzzles.

Using Them
So far we have just used "a" and "b", but they could be anything.

Example: (y+1)2

We can use the (a+b)2 case where "a" is y, and "b" is 1:

(y+1)2 = (y)2 + 2(y)(1) + (1)2 = y2 + 2y + 1

Example: (3x-4)2

We can use the (a-b)2 case where "a" is 3x, and "b" is 4:

(3x-4)2 = (3x)2 - 2(3x)(4) + (4)2 = 9x2 - 24x + 16

Example: (4y+2)(4y-2)

We know that the result will be the difference of two squares, because:

(a+b)(a-b) = a2 - b2

so:

(4y+2)(4y-2) = (4y)2 - (2)2 = 16y2 - 4

Sometimes you can recognize the pattern of the answer:

Example: can you work out which binomials to multiply to get 4x2 - 9

Hmmm... is that the difference of two squares?

Yes! **4x2** is **(2x)2**, and **9** is **(3)2**, so we have:

4x2 - 9 = (2x)2 - (3)2

And that can be produced by the difference of squares formula:

(a+b)(a-b) = a2 - b2

Like this ("a" is 2x, and "b" is 3):

(2x+3)(2x-3) = (2x)2 - (3)2 = 4x2 - 9

So the answer is that you can multiply **(2x+3)** and **(2x-3)** to get **4x2 - 9**

Posted on Aug 09, 2011

SOURCE: definition of special product types and

In mathematics, **special products** are of the form:

(a+b)(a-b) = a2 - b2 (Product of sum and difference of two terms)
which can be used to quickly solve multiplication such as:

301 * 299 = (300 +1)(300-1) = 3002 - 12 = 90000 - 1 = 89999** types**1. Square of a binomial

(a+b)^2 = a^2 + 2ab + b^2

carry the signs as you solve

2. Square of a Trinomial

(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc

carry the sings as you solve

3. Cube of a Binomial

(a+b)^3 = a^3 + 3(a^2)b + 3a(b^2) + b^3

4. Product of sum and difference

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

5. Product of a binomial and a special multinomial

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

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

Posted on Aug 18, 2011

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Posted on Jan 02, 2017

It is a result of a multiplication which follows one of several special laws. These cover many common situations in algebra and maths.

Read more

http://www.intmath.com/factoring-fractions/1-special-products.php

Read more

http://www.intmath.com/factoring-fractions/1-special-products.php

Jun 02, 2015 | Computers & Internet

By definition, the reciprocal of a number is the number divided into one. Thus the product of a number and its reciprocal is one (1).

Oct 06, 2014 | The Learning Company Achieve! Math &...

The common definition is a number only divisible by 1 and itself. This has redundancy for 1, and does not apply.

More formally,**The Fundamental Theorem of Arithmetic**
Every positive integer greater than one can be written *uniquely* as a product of primes, with the prime factors in the product written in order of nondecreasing size.

So 1 is left out.

More formally,

So 1 is left out.

Jun 06, 2014 | Computers & Internet

No need to coin a new word: Polynomial is the accepted form. Review the definition of binomial and polynomial, Choose one indeterminate (variable) X or Y, and create the entities (aX+b)* (cX^5+dX^3+ ....+ 25)

Jun 06, 2014 | Activision Computers & Internet

In algebra there are some identities (that are true all the time)

**(a+b)^2 is always equal to a^2+2ab+b^2**, regardless of the values of a and b. Another one **(a-b)^2=a^2-2ab+b^2**, and

**a^2-b^2=(a-b)(a+b)**

There is also the binomial identity**(a+b)^n.** Look this one up.

There is also the binomial identity

Jun 05, 2014 | Computers & Internet

The product of a sum and difference is the difference of two squares. ie (a+b)*(a-b)=a**2 - b**2 [a squared minus b squared]

Jun 03, 2014 | Activision Computers & Internet

I worked out the algebra leaving out a few simple steps (special binomial products, Pythagorean theorem in trigonometry, definition of secant and co-secant functions). I am inserting the answer as a set of png pictures. However I am not sure how it will look.

Sep 21, 2013 | ValuSoft Bible Collection (10281) for PC

In mathematics, **special products** are of the form:

(a+b)(a-b) = a2 - b2 (Product of sum and difference of two terms) which can be used to quickly solve multiplication such as:

301 * 299 = (300 +1)(300-1) = 3002 - 12 = 90000 - 1 = 89999__types__

1. Square of a binomial

(a+b)^2 = a^2 + 2ab + b^2

carry the signs as you solve

2. Square of a Trinomial

(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc

carry the sings as you solve

3. Cube of a Binomial

(a+b)^3 = a^3 + 3(a^2)b + 3a(b^2) + b^3

4. Product of sum and difference

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

5. Product of a binomial and a special multinomial

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

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

(a+b)(a-b) = a2 - b2 (Product of sum and difference of two terms) which can be used to quickly solve multiplication such as:

301 * 299 = (300 +1)(300-1) = 3002 - 12 = 90000 - 1 = 89999

(a+b)^2 = a^2 + 2ab + b^2

carry the signs as you solve

2. Square of a Trinomial

(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc

carry the sings as you solve

3. Cube of a Binomial

(a+b)^3 = a^3 + 3(a^2)b + 3a(b^2) + b^3

4. Product of sum and difference

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

5. Product of a binomial and a special multinomial

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

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

Jul 28, 2011 | Computers & Internet

In Algebra

Likewise when you see

Special Binomial Products So when you multiply binomials you get ... Binomial Products

And we are going to look at

1. Multiplying a Binomial by Itself What happens when you square a binomial (in other words, multiply it by itself) .. ?

(a+b)2 = (a+b)(a+b) = ... ?

The result:

(a+b)2 = a2 + 2ab + b2

You can easily see why it works, in this diagram:

2. Subtract Times Subtract
And what happens if you square a binomial with a **minus** inside?

(a-b)2 = (a-b)(a-b) = ... ?

The result:

(a-b)2 = a2 - 2ab + b2

3. Add Times Subtract
And then there is one more special case... what if you multiply (a+b) by (a-b) ?

(a+b)(a-b) = ... ?

The result:

(a+b)(a-b) = a2 - b2

That was interesting! It ended up very simple.

And it is called the "**difference of two squares**" (the two squares are **a2** and **b2**).

This illustration may help you see why it works:

a2 - b2 is equal to (a+b)(a-b)
Note: it does not matter if (a-b) comes first:

(a-b)(a+b) = a2 - b2

The Three Cases
Here are the three results we just got:

(a+b)2
= a2 + 2ab + b2
} (the "perfect square trinomials")
(a-b)2
= a2 - 2ab + b2
(a+b)(a-b)
= a2 - b2
(the "difference of squares")
Remember those patterns, they will save you time and help you solve many algebra puzzles.

Using Them
So far we have just used "a" and "b", but they could be anything.

Example: (y+1)2

We can use the (a+b)2 case where "a" is y, and "b" is 1:

(y+1)2 = (y)2 + 2(y)(1) + (1)2 = y2 + 2y + 1

Example: (3x-4)2

We can use the (a-b)2 case where "a" is 3x, and "b" is 4:

(3x-4)2 = (3x)2 - 2(3x)(4) + (4)2 = 9x2 - 24x + 16

Example: (4y+2)(4y-2)

We know that the result will be the difference of two squares, because:

(a+b)(a-b) = a2 - b2

so:

(4y+2)(4y-2) = (4y)2 - (2)2 = 16y2 - 4

Sometimes you can recognize the pattern of the answer:

Example: can you work out which binomials to multiply to get 4x2 - 9

Hmmm... is that the difference of two squares?

Yes! **4x2** is **(2x)2**, and **9** is **(3)2**, so we have:

4x2 - 9 = (2x)2 - (3)2

And that can be produced by the difference of squares formula:

(a+b)(a-b) = a2 - b2

Like this ("a" is 2x, and "b" is 3):

(2x+3)(2x-3) = (2x)2 - (3)2 = 4x2 - 9

So the answer is that you can multiply **(2x+3)** and **(2x-3)** to get **4x2 - 9**

Jul 26, 2011 | Computers & Internet

Here, We deal with Some Special Products in Polynomials.

Certain products of Polynomials occur more often

in Algebra. They are to be considered specially.

These are to be remembered as Formulas in Algebra.

Remembering these formulas in Algebra is as important

as remembering multiplication tables in Arithmetic.

We give a list of these Formulas and Apply

them to solve a Number of problems.

We give Links to other Formulas in Algebra.

Here is the list of Formulas in

Polynomials which are very useful in Algebra.

Formulas in Polynomials :

** Algebra Formula 1 in Polynomials: ** * Square of Sum of Two Terms: *

** (a + b)2 = a2 + 2ab + b2 **

** (a - b)2 = a2 - 2ab + b2 **

** (a + b)(a - b) = a2 - b2 **

** (a + b)(a2 - ab + b2) = a3 + b3 **

** (a - b)3 = a3 - 3a2b + 3ab2 - b3 = a3 - 3ab(a - b) - b3 **

(First term + Second term)2

= (First term)2 + 2(First term)(Second term) + (Second term)2

Jul 02, 2011 | Computers & Internet

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