Nested Form
We have been working with polynomial functions of the form P(x)a_{n}x^{n} + a_{n1}x^{n1} + ^{ ... } + a_{2}x^{2} + a_{1}x + a_{0}. We can also write polynomials in
nested form. The nested form of a polynomial is:
P(x) = (((((a)x + b)x + c)x + d )x + ^{ ... })
The nested form is useful when evaluating a polynomial function by hand.
Here are the steps to converting a polynomial into nested form:
 Write the polynomial in descending order
 Factor x out of all the terms in which it appears.
 Factor x out of all the terms in parentheses in which it appears.
 Repeat step 3 until only a constant remains in the innermost parentheses.
Example 1: Convert P(x) = 6x^{2} 7x + 3x^{4} +11  2x^{3} to nested form.
P(x)  =  3x^{4} 2x^{3} +6x^{2}  7x + 11 

 =  (3x^{3} 2x^{2} + 6x  7)x + 11 

 =  ((3x^{2}  2x + 6)x  7)x + 11 

 =  (((3x  2)x + 6)x  7)x + 11 

 =  ((((3)x  2)x + 6)x  7)x + 11. 

Nested form allows for easy evaluation of a polynomial without a calculator.
For example, P(3) = ((((3)3  2)3 + 6)3  7)3 + 11 = (((7)3 + 6)3  7)3 + 11 = ((27)3  7)3 + 11 = (74)3 + 11 = 233.
Example 2: Convert P(x) =  8x^{3} +7x  8x^{4} +2x^{5}  x^{2} + 3 to nested
form and evaluate P(5).
P(x)  =  2x^{5} 8x^{4} 8x^{3}  x^{2} + 7x + 3 

 =  (2x^{4} 8x^{3} 8x^{2}  x + 7)x + 3 

 =  ((2x^{3} 8x^{2}  8x  1)x + 7)x + 3 

 =  (((2x^{2}  8x  8)x  1)x + 7)x + 3 

 =  ((((2x  8)x  8)x  1)x + 7)x + 3 

 =  (((((2)x  8)x  8)x  1)x + 7)x + 3. 

P(5) = (((((2)5  8)5  8)5  1)5 + 7)5 + 3 = ((((2)5  8)5  1)5 + 7)5 + 3 = (((2)5  1)5 + 7)5 + 3 = ((9)5 + 7)5 + 3 = (52)5 + 3 = 263.