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Definition of special products

Can u help me with definition of special products?

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

Posted on Feb 09, 2011

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Special product is RTOOOOT!! :)

Posted on Aug 05, 2012

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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:

x-y-2-diagram.gif
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:
apb-amb-why.gif 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

  • 132 Answers

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 = 89999types
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

6ya6ya
  • 2 Answers

SOURCE: I have freestanding Series 8 dishwasher. Lately during the filling cycle water hammer is occurring. How can this be resolved

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Posted on Jan 02, 2017

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Definition of special product types and examples of special products


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 = 89999types
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

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Definition of special product in algebra types and example of special product in algebra


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:

x-y-2-diagram.gif
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:
apb-amb-why.gif 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

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