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Nested Form

We have been working with polynomial functions of the form P(x)anxn + an-1xn-1 + ... + a2x2 + a1x + a0. 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:

  1. Write the polynomial in descending order
  2. Factor x out of all the terms in which it appears.
  3. Factor x out of all the terms in parentheses in which it appears.
  4. Repeat step 3 until only a constant remains in the innermost parentheses.


Example 1: Convert P(x) = 6x2 -7x + 3x4 +11 - 2x3 to nested form.


P(x)=3x4 -2x3 +6x2 - 7x + 11  
 =(3x3 -2x2 + 6x - 7)x + 11  
 =((3x2 - 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) = - 8x3 +7x - 8x4 +2x5 - x2 + 3 to nested form and evaluate P(5).


P(x)=2x5 -8x4 -8x3 - x2 + 7x + 3  
 =(2x4 -8x3 -8x2 - x + 7)x + 3  
 =((2x3 -8x2 - 8x - 1)x + 7)x + 3  
 =(((2x2 - 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.

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