(a+b)^3 as a polynomial in standard form

Question

Just standard form

Answer 1

Method 1: Horizontal Multiplication                                          

 

Step 1 : Expand

 

(a+b)³ = (a+b)(a+b)(a+b)    

 

Step 2: Multiply the first two binomials together. Multiply each term in the first parenthesis with each in the second, (Front terms, Outside term, Inner term, Last terms, also known as FOILING).
                                               

 

           = a*a + a*b + b*a +b*b (a+b)    

 

Step 3:   Combine like terms to simplify the new front polynomial.

                            

           = (a² + 2ab +b²) (a+b)    

 

Step 4:  Multiply each term from the front polynomial to each term in the back binomial.
                               

              =  a²*a + 2ab*a +b²*a +a²* b + 2ab²*b + b²*b

 

Step 5: Combine like terms and then order terms alphabetically with highest power for a to the left and least power of a to the far right.

 

        

 Answer =   a³+ 3a²b +3ab² + b³          

 

                                            

Method 2:  Shortcut using Pascal's Triangle

 

 Use the numbers from the 3rd row of Pascal's Triangle

 

 (a+b)³ →  1 3 3 1                                                            

 

Pascal's triangle

         1

      1    1

    1   2    1

  1   3   3   1

1   4   6   4   1

               

 

Step 2:  Fill in accordingly, the first and last term with the 1 coefficients, then fill in the middle terms alphabetically with the powers descending left to right. All terms must have a sum for the exponents of the variables add up to 3 which is the power of the original binomial.

 

1a³ + 3a²b¹ + 3a¹b² + 1b³             

 

 

Step 3:  Simplify. It is not necessary to write a 1 for coefficients or exponents. It is assumed that they are there.                    

 

 Answer = a³+ 3a²b +3ab² + b³                                         

 

Answer 2

First, we expand the binomial based on the power. 

 

(a + b)(a + b)(a + b)

 

Next, we expand this into a polynomial.  This means that the first term of the polynomial will be a³ and the last term will be b³.  The best way to do this is to use FOIL method on the first two binomials.  Then we distribute the last binomial into the product.

 

 

(a² + 2ab + b²) (a + b) =

 

a(a² + 2ab + b²) + b(a² + 2ab + b²)

 

Think of the polynomial  a² + 2ab + b²  as the greatest common factor in the distributive property.  Next, is to distribute terms.

 

a³ + 2a²b + ab² + a²b + 2ab² + b³

 

 

Combine like terms.

 

a³ + 3a²b + 3ab² + b³