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:
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. |