How to evaluate algebraic expressions mentally

If f(x) = 5x^4 – 26x^3 – 27x^2 + 17x + 3, then f(6) = ?

This problem is from David Robinson’s “Mental Math Challenge for Teachers”. Robinson is a retired teacher who handled various high school Math subjects for 33 years.

The conventional solution is to substitute 6 for x in the expression, calculate each term separately and combine the numerical results:

Using a calculator, we have

5x^4        = 5(6^4)     = 5(1296)      = 6480

– 26x^3   = -26(6^3) = – 26(216) = – 5616  

 – 27x^2  = -27(6^2) = – 27 (36)  =  – 972

+ 17x        = 17 (6)                          =    102      

+ 3                                                              3

6480 – 5616 – 972 + 102 + 3 = -3

Now according to Mr. Robinson, that expression can be evaluated mentally in 18 seconds. Here’s how it is done:

Multiply the coefficient of the first term(5)  by the value of x(6) and add the coefficient of the next term(-26).

Repeat:

     Multiply the result of the previous step by 6 and add the coefficient of the next

Until the end

5(6) – 26 = 30 – 26 = 4

4(6) – 27 = 24 – 27 = -3

-3(6) + 17 = – 18 + 17 = -1

-1(6) + 3 = -6 + 3 = -3