Wednesday, December 20, 2017

Middle School Math Solutions – Expand Calculator, Perfect Cube

Similarly to the perfect square formula, which we covered in the last post, we get the perfect cube formula. The perfect cube is when we cube a binomial. We use polynomial multiplication and the FOIL method for this formula.

Let’s take a look at it:

                                                    (a+b)^3=a^3+3a^2 b+3ab^2+b^3

Here’s the proof:

                                                    (a+b)^3=(a+b)(a+b)(a+b)
                                                                   =(a+b)(a^2+2ab+b^2)
                                                                   =a(a^2+2ab+b^2 )+b(a^2+2ab+b^2)
                                                                   =a^3+2a^2 b+ab^2+a^2 b+2ab^2+b^3
                                                                   =a^3+3a^2 b+3ab^2+b^3

As you can see there are many steps involved, so having this formula will help you solve perfect cube problems quickly.

Let’s see some examples.

First example (click here):

                                                                  Expand (x+1)^3

 1.  Apply the formula

                                                                       a=x,b=1

                                                 (x+1)^3=x^3+3∙x^2∙1+3∙x∙1^2+1^3
                                                                =x^3+3x^2+3x+1 

Next example (click here):

                                                                 Expand (2x-2)^3

1. Apply the formula

                                                                    a=2x,b=-2

                               (2x-2)^3=(2x)^3+3∙(2x)^2∙(-2)+3∙(2x)∙(-2)^2+(-2)^3
                                                =8x^3-24x^2+24x-8                             

Last example (click here):

                                                               Expand (s^2+2t)^3

1. Apply the formula

                                                                   a=s^2,b=2t

                                 (s^2+2t)^3=(s^2 )^3+3∙(s^2 )^2∙(2t)+3∙s^2∙(2t)^2+(2t)^3
                                                   =s^6+6s^4 t+12s^2 t^2+8t^3               

Not too difficult! Memorizing this formula will help you and become faster at cubing binomials. If you are looking for more practice problems or help, check out Symbolab’s Practice.

Until next time,

Leah

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