Chapter 13: Q. 5 (page 870)

Find the complex zeros of:
$f (x) = 5 x^{4} - 9 x^{3} - 7 x^{2} - 31 x - 6$

Short Answer

Expert verified

The two complex zeros of the function is: $x = \frac{- 1 \pm i \sqrt{7}}{2}$

Step by step solution

Step 1. Given information

The function is:

$f (x) = 5 x^{4} - 9 x^{3} - 7 x^{2} - 31 x - 6$

Step 2. Draw the graph of the function by using graphing calculator.

The graph of the function is:

From this graph it is clear that one of the real zero of the function is $x = 3$

So $(x - 3)$ will be one factor of $f (x)$ .

Step 3. To find the depressed equation.

Since $f (3) = 0$

So, by using synthetic division:

localid="1647766430444" $3 5 - 9 - 7 - 31 - 6 15 18 33 6 5 6 11 2 0$

So, $f (x) = (x - 3) (5 x^{3} + 6 x^{2} + 11 x + 2)$

So depresses equation is:

localid="1647764958561" $5 x^{3} + 6 x^{2} + 11 x + 2 = 0$

The one of the real root of this depressed equation is $x = - \frac{1}{5} x = - 0.2$

Step 4. To factorize depressed equation.

$- 0.2 5 6 11 2 - 1 - 1 - 2 5 5 10 0$

The depressed equation can be factored as:

role="math" localid="1647766777287" $(x + 0.2) (5 x^{2} + 5 x + 10)$

Step 5. Solve the quadratic equation.

$5 x^{2} + 5 x + 10 = 0$

$x = \frac{- 5 \pm \sqrt{5^{2} - 4.5.10}}{2.5} x = \frac{- 5 \pm \sqrt{25 - 200}}{10} x = \frac{- 5 \pm \sqrt{- 175}}{10} x = \frac{- 5 \pm \sqrt{175 i^{2}}}{10} x = \frac{- 5 \pm i 5 \sqrt{7}}{10} x = \frac{5 (- 1 \pm i \sqrt{7)}}{10} x = \frac{- 1 \pm i \sqrt{7}}{2}$

So there are two complex zeros of the given function. $x = \frac{- 1 \pm i \sqrt{7}}{2}$

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Find the complex zeros of:
$f (x) = 5 x^{4} - 9 x^{3} - 7 x^{2} - 31 x - 6$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Draw the graph of the function by using graphing calculator.

Step 3. To find the depressed equation.

Step 4. To factorize depressed equation.

Step 5. Solve the quadratic equation.

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Find the complex zeros of:f(x)=5x4-9x3-7x2-31x-6

Short Answer

Step by step solution

Step 1. Given information

Step 2. Draw the graph of the function by using graphing calculator.

Step 3. To find the depressed equation.

Step 4. To factorize depressed equation.

Step 5. Solve the quadratic equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Discrete Mathematics

Theoretical and Mathematical Physics

Logic and Functions

Geometry

Study anywhere. Anytime. Across all devices.

Find the complex zeros of:
$f (x) = 5 x^{4} - 9 x^{3} - 7 x^{2} - 31 x - 6$