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:
- 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) = 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.
