Differentiate the following polynomials.Note: some questions may not include all the possible terms.-x5 – 5x4 + 5³ –11æ? – 21x – 40f'(x) =f" (x) = |

Answered: -25 – 5x4 + 5x³ – 11æ² – 21x – 40

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Complex Numbers

Section4.2: Complex Solutions Of Equations

Problem 7ECP: Find the cubic polynomial function f with real coefficients that has 1 and 2+i as zeros, and f2=2.

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Expert Solution

Step 1

Given:

$f (x) = - x^{5} - 5 x^{4} + 5 x^{3} - 11 x^{2} - 21 x - 40$

To find:

$f' (x)$ and $f'' (x)$ .

Step 2

Differentiate f(x) with respect to x it yields f'(x),

Thus,

$\frac{d}{d x} f (x) = \frac{d}{d x} (- x^{5} - 5 x^{4} + 5 x^{3} - 11 x^{2} - 21 x - 40)$

$f' (x) = \frac{d}{d x} (- x^{5} - 5 x^{4} + 5 x^{3} - 11 x^{2} - 21 x - 40)$

$f' (x) = \frac{d}{d x} (- x^{5}) + \frac{d}{d x} (- 5 x^{4}) + \frac{d}{d x} (5 x^{3}) - \frac{d}{d x} (11 x^{2}) - \frac{d}{d x} (21 x) - \frac{d}{d x} (40)$

Using the power rule $\frac{d}{d x} x^{n} = n x^{n - 1}$ ,

Thus, above equation become,

$f' (x) = (- 5 x^{5 - 1} - 5 (4) x^{4 - 1} + 5 (3) x^{3 - 1} - 11 (2) x^{2 - 1} - 21)$

On simplifying,

$f' (x) = (- 5 x^{4} - 5 (4) x^{3} + 5 (3) x^{2} - 11 (2) x^{1} - 21)$

f' (x) = (- 5 x^{4} - 20 x^{3} + 15 x^{2} - 22 x^{1} - 21)

Which is required.

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