Chapter 4: Q.30 (page 215)

In Problems 23–30, use the given zero to find the remaining zeros of each function.
$g (x) = 2 x^{5} - 3 x^{4} - 5 x^{3} - 15 x^{2} - 207 x + 108; zero : 3 i$

Short Answer

Expert verified

The remaining answer is $\pm 3 i, - \frac{1}{2}, 3, 4 .$

Step by step solution

Step 1. Given Information.

The given polynomial is

$g (x) = 2 x^{5} - 3 x^{4} - 5 x^{3} - 15 x^{2} - 207 x + 108$

Step 2. Performing Long Division.

Here we apply the conjugate the Pairs Theorem. The degree of the polynomial is 5.

3i is zero of polynomial then it's conjugate -3i also zero of the polynomial. Now we have two zeros.

Since all the coefficients are $g (x)$ are real and $3 i$ is a root of $g (x)$ then $- 3 i$ is also a root of $g (x)$ . Then $(x + 3 i) (x - 3 i)$ are factors of $g (x)$ . So by divides $(x - 3 i) (x + 3 i) = x^{2} + 9$ to $g (x)$ .

$\frac{(2 x^{5} - 3 x^{4} - 5 x^{3} - 15 x^{2} - 207 x + 108)}{(x^{2} + 9)} = 2 x^{3} - 3 x^{2} - 23 x + 12$

Step 3. Factors of the finding polynomial.

Now, find the factors and apply the zero methods.

2 x^{3} - 3 x^{2} - 23 x + 12 = 2 x^{2} (x - 4) - 5 x (x - 4) - 3 (x - 4)

And,

$\begin{array}{rcl} (x - 4) (2 x^{2} - 5 x - 3) & = & (x - 4) (2 x^{2} - 6 x + x - 3) \\ = & (x - 4) {2 x (x - 3) + 1 (x - 3)} \\ = & (x - 4) (2 x + 1) (x - 3) \end{array}$

The zeros are $\pm 3 i, - \frac{1}{2}, 3, 4 .$

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In Problems 23–30, use the given zero to find the remaining zeros of each function.
$g (x) = 2 x^{5} - 3 x^{4} - 5 x^{3} - 15 x^{2} - 207 x + 108; zero : 3 i$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Performing Long Division.

Step 3. Factors of the finding polynomial.

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In Problems 23–30, use the given zero to find the remaining zeros of each function.gx=2x5-3x4-5x3-15x2-207x+108;zero:3i

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Performing Long Division.

Step 3. Factors of the finding polynomial.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Theoretical and Mathematical Physics

Geometry

Statistics

Logic and Functions

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 23–30, use the given zero to find the remaining zeros of each function.
$g (x) = 2 x^{5} - 3 x^{4} - 5 x^{3} - 15 x^{2} - 207 x + 108; zero : 3 i$