Special Products - Algebra

 Special Products

            - are formulas that makes solving for products of some algebraic expression easier.

* These are the followings:

1. Product of Two Binomials
           

            - solved by using FOIL method.

F - irst to first

O - utside to outside

I - nside to inside

L - ast to last

Formula:

(ax + by)(cx + dy) = acx² + (ad + bc)xy + bdy²

Where:

x and y are variables
a, b, c, and d are numerical coefficients

Note:
- Start with the first term of one of the binomials distribute it with the other next proceed with the second term of the binomial where you get the first term then continue the distribution.
- Same method will be applied with a binomial with one variable or its second term is a constant.

Examples:

* (5x + 3)(2x - 3) = (5x)(2x) + (5x)(-3) + (3)(2x) + (3)(-3)

= 10x² - 15x + 6x - 9

= 10x² - 9x - 9

* (2x + 3y)(5x + y) = (2x)(5x) + (2x)(y) + (3y)(5x) + (3y)(y)

= 10x² + 2xy + 15xy + 3y²

= 10x² + 17xy + 3y²

2. Square of Binomial
           

            - The square of a binomial is a perfect square trinomial

Formula:

(x + y) ² = x² + 2xy + y²

(x - y)² = x² - 2xy + y²

Examples:

* (x - 3)² = x² - 2x(3) + 9

= x² - 6x + 9

* (x² + 2)² = x⁴ + 2x²(2) + 4

= x⁴ + 4x² + 4

3. Product of Sum and Difference of Binomials

Formula:

(x + y)(x - y) = x² – y²

Examples:

* (5x + 3)(5x - 3)

= 25x² - 9

* (3x¹⁰ + 9x⁹)(3x¹⁰ - 9x⁹)

= 9x²⁰ - 81x¹⁸

4. Cube of Binomial

Formula:

(x + y)³ = x³ + 3x²y + 3xy² + y³
(x - y)³ = x³ - 3x²y + 3xy² - y³

Example:

* (2x + 4)³

= (2x) ³ + 3(2x) ²(4) + 3(2x) (4)² + (4)³

= 8x³ + 48x² + 96x + 64

5. Special Case of Product of Binomial and Trinomial

Formula:

(x + y)(x² - xy + y²) = x³ + y³
(x - y)(x² + xy + y²) = x³ + y³

Example:

(2x + 3)(4x² - 6x + 9)

= 8x³ + 27

6. Square of Trinomial

Formula:

(a + b + c) ² = a² + b² + c² + 2ab + 2ac + 2bc

Example:

(2x + 3y + z)²

= (2x)² + (3y)² + (z)² + 2(2x)(3y) + 2(2x)(z) + 2(3y)(z)

= 4x² + 9y² + z² + 2(2x)(3y) + 2(2x)(z) + 2(3y)(z)

= 4x² + 9y² + z² + 12xy + 4xz + 6yz