In this section we * learn the nested scheme*, which is also known as

So, for instance, by the end of this section we'll be able to calculate \(f(x) = x^5 - 4x^3 + 2x^2 - x + 3\) when \(x = 3\), without a calculator, with a quick *algorithm* and show that \(f(3)=153\).

We *learn the method* in the *tutorial* below and then practice with some exercise questions.

To learn the *method* we work through two examples. In the first we work through a typical introductory example in which we evaluate \(f(x) = 2x^4 - 3x^3 +x^2 - 2x+3\) when \(x = 2\).

In the second example we learn how to deal with cases in which ** some of the coefficients of the polynomial are equal to zero** by evaluating \(f(x) = -x^5 + 2x^3 + 3x + 5\) when \(x = -2\), in which the \(x^4\) and the \(x^2\) coefficients are equal to \(0\).

Using the *nested scheme for evaluating polynomials*, answer each of the following:

- Evaluate \(f(x) = -x^2+3x - 9\) when \(x = 7\).
- Evaluate \(f(x) = -3x^5 + x^4 - 2x^3 + 6x^2 + 2x - 8\) when \(x = 2\).
- Evaluate \(f(x) = -2x^7 + 3x^5 + 2x^4 - 4x + 1\) when \(x = -2\).
- Evaluate \(f(x) = x^5 - 4x^3 + 2x^2 - x +3 \) when \(x = 3\).
- Evaluate \(f(x) = -x^6 + 2x^5 - 3x^3 + x^2 - 4x + 5\) when \(x = -2\).
- Evaluate \(f(x) = -8x^4 + 6x^3 + 2x^2 - 3x + 1\) when \(x = \frac{1}{2}\).
- Evaluate \(f(x) = 2x^4 - 7x^3 + 8x^2 + 6\) when \(x = \frac{3}{2}\).
- Evaluate \(f(x) = -27x^4 + 9x^2 - 4x + 5\) when \(x = -\frac{2}{3}\).

- For \(f(x) = -x^2+3x - 9\) when \(x = 7\), we find: \[f(7) = -37\]
- For \(f(x) = -3x^5 + x^4 - 2x^3 + 6x^2 + 2x - 8\) when \(x = 2\), we find: \[f(2) = -76\]
- For \(f(x) = -2x^7 + 3x^5 + 2x^4 - 4x + 1\) when \(x = -2\), we find: \[f(-2) = 201\]
- For \(f(x) = x^5 - 4x^3 + 2x^2 - x +3 \) when \(x = 3\), we find: \[f(3) = 153\]
- For \(f(x) = -x^6 + 2x^5 - 3x^3 + x^2 - 4x + 5\) when \(x = -2\), we find: \[f(-2) = -87\]
- For \(f(x) = -8x^4 + 6x^3 + 2x^2 - 3x + 1\) when \(x = \frac{1}{2}\), we find: \[f\begin{pmatrix}\frac{1}{2}\end{pmatrix} = \frac{1}{4} = 0.25\]
- For \(f(x) = 2x^4 - 7x^3 + 8x^2 + 6\) when \(x = \frac{3}{2}\), we find: \[f\begin{pmatrix}\frac{3}{2} \end{pmatrix} = \frac{21}{2} = 10.5\]
- For \(f(x) = -27x^4 + 9x^2 - 4x + 5\) when \(x = -\frac{2}{3}\), we find: \[f\begin{pmatrix}-\frac{2}{3}\end{pmatrix} = \frac{19}{3}\]

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