What makes these products "special"?
On this page, you will find algebraic products you will use repeatedly in this chapter and in a lot of math you will encounter later on.They are "special" because they are very common, and they are worth knowing about.
It is easier later on if you can recognize these products easily.
Special Products involving Squares
Multiplying out brackets creates the following special products.Knowing these well is worth knowing because you'll need them often.
a(x + y) = ax + ay (Distributive Law)
(x + y)(x − y) = x2 − y2 (Difference of 2 squares)
(x + y)2 = x2 + 2xy + y2 (Square of a sum)
(x − y)2 = x2 − 2xy + y2 (Square of a difference)
Examples using the special products
In Example 1, multiply out 2*(a − 3)
Provide an answer
This product utilizes the first one above.We multiply the term outside the bracket (the "2x") by the terms within the brackets (the "a" and "−3").
2x(a − 3) = 2ax − 6x
We recognize this one involves the Difference of 2 squares:
(7s + 2t)(7s − 2t)
= (7s)2 − (2t)2
= 49s2 − 4t2
(12 + 5ab)(12 − 5ab)
= (12)2 − (5ab)2
= 144 − 25a2b2
The answer is a difference of 2 squares.
This one is the square of a sum of 2 terms.
(5a + 2b)2
= (5a)2 + 2(5a)(2b) + (2b)2
= 25a2 + 20ab + 4b2
(q − 6)2
= (q)2 − 2(q)(6) + (6)2
= q2 − 12q + 36
This example involved the square of a difference of 2 terms.
This question is not in any of the formats we have above. So we just need to multiply out the brackets, term-by-term.
It"s important to recognize when we have a Special Product and when our question is something else.
(8x − y)(3x + 4y)
= 8x(3x + 4y) − y(3x + 4y)
= 8x(3x) + 8x(4y) − y(3x) − y(4y)
= 24x2 + 32xy −3xy − 4y2
= 24x2 + 29xy − 4y2
To expand this, we put it in the form (a + b)2 and expand it using the third rule above, which says:
(a + b)2 = a2 + 2ab + b2
I put
a = x + 2
b = 3y
This gives me:
(x + 2 + 3y)2
= (<x + 2> + 3y)2
= <x + 2>2 + 2<x + 2>(3y) + (3y)2
= <x2 + 4x + 4> + (2x + 4)(3y) + 9y2
= x2 + 4x + 4 + 6xy + 12y + 9y2
I could have chosen the following and obtained the same answer:
a = x
b = 2 + 3y
Try it!
Special Products involving Cubes
The following products are just the result of multiplying out the brackets.
(x + y)3 = x3 + 3x2y + 3xy2 + y3 (Cube of a sum)
(x − y)3 = x3 − 3x2y + 3xy2 − y3 (Cube of a difference)
(x + y)(x2 − xy + y2) = x3 + y3 (Sum of 2 cubes)
(x − y)(x2 + xy + y2) = x3 − y3 (Difference of 2 cubes)
These are also worth knowing well enough so you recognize the form, and the differences between each of them. (Why? Because it"s easier than multiplying out the brackets and it helps us solve more complex algebra problems later.)
Example 8: Expand(2s + 3)3
Answer
This involves the Cube of a Sum:
(2s + 3)3
= (2s)3 + 3(2s)2(3) + 3(2s)(3)2 + (3)3
= 8s3 + 36s2 + 54s + 27
Exercises
Expand:
(1) (s + 2t)(s − 2t)
Answer
Using the Difference of 2 Squares formula
(x + y)(x − y) = x2 − y2,
we have:
(s + 2t)(s − 2t)
= (s)2 − (2t)2
= s2 − 4t2
(2) (i1 + 3)2
Answer
Using the Square of a Sum formula
(x + y)2 = x2 + 2xy + y2,
we have:
(i1 + 3)2
= (i1)2 + 2(i1)(3) + (3)2
= i12 + 6i1 + 9
(3) (3x + 10y)2
Answer
Using the Square of a Sum formula
(x + y)2 = x2 + 2xy + y2,
we have:
(3x + 10y)2
= (3x)2 + (2)(3x)(10y) + (10y)2
= 9x2 + 60xy + 100y2
(4) (3p − 4q)2
Answer
Using the Square of a Difference formula
(x − y)2 = x2 − 2xy + y2,
we have:
(3p − 4q)2
= (3p)2 − (2)(3p)(4q) + (4q)2
= 9p2 − 24pq + 16q2
top
Factoring and Fractions
2. Common Factor and Difference of Squares
Related, useful or interesting posturasdeyogafaciles.com articles
Ten Ways to Survive the Math Blues
Fear math? Here are 10 useful tips to survive your next math class. Read more »
Should math be applied?
Some research from Ohio State University concludes that we should teach mathematics without the use of "real life" concrete examples.I beg to differ. Read more »
posturasdeyogafaciles.com forum
Latest Factoring and Fractions forum posts:Got questions about this chapter?
Fraction algebra by phinah
Lowest common denominator by John
Equivalent Fractions by Taradawn
Factoring by Michael
Equivalent fractions question by Michael
Search posturasdeyogafaciles.com
Search posturasdeyogafaciles.com
Search:
Click to search:
Online Algebra Solver
This algebra solver can solve a wide range of math problems.
Go to: Online algebra solver
Subscribe
* indicates required
Email Address *
Name (optional)
Home | Sitemap |Author: Murray Bourne |About & Contact |Privacy & Cookies | posturasdeyogafaciles.com feed |Page last modified: 19 February 2018