N2 + 18n + 81 need help

(n + 9)^2

Explanation:

Step  1  :

Trying to factor by splitting the middle term

1.1     Factoring  n2+18n+81

The first term is,  n2  its coefficient is  1 .

The middle term is,  +18n  its coefficient is  18 .

The last term, "the constant", is  +81

Step-1 : Multiply the coefficient of the first term by the constant   1 • 81 = 81

Step-2 : Find two factors of  81  whose sum equals the coefficient of the middle term, which is   18 .

     -81    +    -1    =    -82

     -27    +    -3    =    -30

     -9    +    -9    =    -18

     -3    +    -27    =    -30

     -1    +    -81    =    -82

     1    +    81    =    82

     3    +    27    =    30

     9    +    9    =    18    That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  9  and  9

                    n2 + 9n + 9n + 81

Step-4 : Add up the first 2 terms, pulling out like factors :

                   n • (n+9)

             Add up the last 2 terms, pulling out common factors :

                   9 • (n+9)

Step-5 : Add up the four terms of step 4 :

                   (n+9)  •  (n+9)

            Which is the desired factorization

Multiplying Exponential Expressions :

1.2    Multiply  (n+9)  by  (n+9)

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is  (n+9)  and the exponents are :

         1 , as  (n+9)  is the same number as  (n+9)1

and   1 , as  (n+9)  is the same number as  (n+9)1

The product is therefore,  (n+9)(1+1) = (n+9)2

Final result :

 (n + 9)^2

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