Difference of Squares, Square of a Binomial

Question

Factor:

50x^2−72y^2 =

Answer 1

50x² − 72y² = ?

A binomial in the form a² – b² can be factored to give (a + b)(a – b).

Our starting binomial is not in this form, but that can be fixed...

Since the x² and y² coefficient are both even we there is a common factor of two with allows us unveil the relationship we are looking for:

50x² − 72y² = 2(25x² − 36y²)

25x² = (5x)² and 36y² = (6y)² which means

25x² − 36y² = (5x)² − (6y)² = (5x + 6y)(5x – 6y)

Answer

50x² − 72y² = 2(5x + 6y)(5x – 6y)

Answer 2

The first step is always to remove the largest possible monomial factor. Since x^2 and y^2 have no common factors, we look at their coefficients. 50 and 72 are both even, so they both have 2 as a factor. Dividing by 2, we get 25 and 36, which have no common factors other than 1. However, they are both perfect squares: 25 = 5 x 5 and 36 = 6 x 6. So we now have 50x^2 - 72y^2 = 2 * (5*5 * x*x - 6*6 * y*y). We now apply the formula for factoring the difference of two squares: a^2 - b^2 = (a + b) * (a - b). Here, a can be 5x and b can be 6y, so 50x^2 - 72y^2 = 2(5x + 6y)(5x - 6y).

Answer 3

50x² - 72y² = 2[25x² - 36y²] = 2[(5x)² - (6y)²] = 2(5x - 6y)(5x + 6y)

Answer 4

First, factor out the greatest common factor:

50x^2−72y^2 = 2 * ( 25x² - 36y² ) = 2 ( 5²x² - 6²y² )

= 2 ( (5x)² - (6y)² ) = 2 ( 5x + 6y ) ( 5x - 6y )

Note:

a²b² = ( ab )²

and

( a² - b² ) = ( a + b ) ( a - b )

IMPORTANT TERMINOLOGY:

A monomial is the product of one or more numbers, variables, and variables and numbers. A term is another word for a monomial. Examples: 2, 3, 5x²y, -12y

Superscripted numbers are called powers or exponents. They always come after a variable; if there is none visible, it is implied to be 1. Exponents are simply shortcuts for showing how many factors they represent: x³ = x × x × x

The number in front of the variable prouct is called the coefficient. If there is none visible, it is implied to be 1. Coefficients tell us how many we have: 3y means you have 3 y's.

If you add or subtract terms (monomial), you create a polynomial. If there are two terms, you have a binomial: 2x + 1, for instance. Three terms is a trinomial: -x² + 6y - 2.