factoring polynomails

Question

1)       3

   125x-1 =

 

2)   2

    w-64=

 

3) 2

   y-12y+36+

 

6)      2    2

    25a+8b=

Answer 1

Hi Amanda.

 

#1 is a difference of cubes.  I tell my students that they must memorize the formula for sums and differences of cubes.

 

The formulas are:

 

a³ - b³ = (a - b) (a² + ab + b²)

 

a³ + b³ = (a + b) (a² - ab + b²)

 

 

#2 is a difference of squares.  I tell my students that they must memorize the formula for the difference of squares.

 

The formula is:

 

a² - b² = (a - b) (a + b)

 

#3 is "prime", unless you are allowed to use complex numbers.  

 

 

So...

 

#1:  125x³ - 1 = (5³)(x³) - 1³ 

         

                      =   (5x)³ - 1³ 

 

This is a difference of cubes so it factors as:

 

(5x - 1)(25x² + 5x +1)      (see formula above)

 

 

#2  w² - 64 = w² - 8²

 

This is a difference of squares so it factors as:

 

(w - 8)(w + 8)

 

 

#3 is prime, and does not factor in the "real" world.

 

 

PLEASE IGNORE THE FOLLOWING if you are not using complex numbers....

 

If you are allowed to use complex numbers, then 

 

25a² + 8b² = (5a)² +[(2√2)(b)]²

 

(5a)² +[(2√2)(b)]² = [(5a) - (2√2)(i)(b)] [(5a) + (2√2)(i)9b)]