Algebra II: Polynomials
Nested Form of a Polynomial
Nested Form
We have been working with polynomial functions of the form
P(x)a
n
x
n + a
n-1
x
n-1 + ... + 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 .
