---A polynomial $ f(x) $ and two of its zeros are given.\[ f(x) = 2x^5 + 11x^4 + 22x^3 - 62x^2 - 120x + 72; \]\[ -3 - 3i \quad \text{and} \quad \frac{1}{2} \quad \text{are zeros} \]---This excerpt presents a polynomial function $ f(x) $ along with two of its zeros. The polynomial is of the fifth degree, given by:\[ f(x) = 2x^5 + 11x^4 + 22x^3 - 62x^2 - 120x + 72; \]Additionally, the zeros of the polynomial provided are:\[ -3 - 3i \quad \text{and} \quad \frac{1}{2} \]In this context, a zero of a polynomial is a value of $ x $ for which $ f(x) = 0 $. **Note for Students**: Remember that if a polynomial has a complex zero $ a + bi $, the conjugate $ a - bi $ is also a zero if the coefficients of the polynomial are real.Understanding these concepts is crucial in solving polynomial equations and finding all the roots (zeros) of the given polynomial function.--- ### Factoring $ f(x) $ as a Product of Linear Factors#### Problem Statement(b) Factor $ f(x) $ as a product of linear factors.\[ f(x) = \]#### ExplanationTo solve this problem, you need to express the function $ f(x) $ as a product of linear factors. A linear factor is generally in the form $(x - a)$, where $a$ is a root of the equation.#### Input Box and Tool MenuIn the provided image, there appears to be an empty input box next to $ f(x) = $. This is where you will input the factored form of the function. Additionally, there is a tool menu with various mathematical symbols:1. **i**: Likely to represent the imaginary unit.2. **Fraction Symbol**: Indicates the option to input fractions.3. **Power Symbol**: Allows you to input exponents.4. **Square Root Symbol**: Permits entry of square roots.#### Graphs or DiagramsThere are no graphs or diagrams associated with this problem statement.#### Steps to Factor $ f(x) $1. Identify the roots of the polynomial $ f(x) $. This could be done through techniques such as solving the equation $ f(x) = 0 $, synthetic division, or factoring by grouping.2. Once the roots $ r_1, r_2, \dots, r_n $ are identified, express $ f(x) $ as:\[ f(x) = a(x - r_1)(x - r_2) \cdots (x - r_n) \]Where $ a $ is the leading coefficient of the polynomial.#### ExampleFor instance, if the polynomial is $ f(x) = x^2 - 5x + 6 $, you can find that the roots are $ x = 2 $ and $ x = 3 $. Thus, the factorized form is:\[ f(x) = (x - 2)(x - 3) \]Remember, the exact factorization depends on the given function $ f(x) $. The provided tools can help ensure correct formatting of mathematical expressions.

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Answered: A polynomial f (x) and two of its zeros…

Math

Advanced Math

A polynomial f (x) and two of its zeros are given. f(x)=2x° +11x* +22x' – 62x² – 120x+72; 1 -3-3i and are zeros 2

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.

Question

---

A polynomial $ f(x) $ and two of its zeros are given.

\[ f(x) = 2x^5 + 11x^4 + 22x^3 - 62x^2 - 120x + 72; \]

\[ -3 - 3i \quad \text{and} \quad \frac{1}{2} \quad \text{are zeros} \]

---

This excerpt presents a polynomial function $ f(x) $ along with two of its zeros. The polynomial is of the fifth degree, given by:

\[ f(x) = 2x^5 + 11x^4 + 22x^3 - 62x^2 - 120x + 72; \]

Additionally, the zeros of the polynomial provided are:

\[ -3 - 3i \quad \text{and} \quad \frac{1}{2} \]

In this context, a zero of a polynomial is a value of $ x $ for which $ f(x) = 0 $.

**Note for Students**: Remember that if a polynomial has a complex zero $ a + bi $, the conjugate $ a - bi $ is also a zero if the coefficients of the polynomial are real.

Understanding these concepts is crucial in solving polynomial equations and finding all the roots (zeros) of the given polynomial function.

---

### Factoring $ f(x) $ as a Product of Linear Factors

#### Problem Statement

(b) Factor $ f(x) $ as a product of linear factors.

\[ f(x) = \]

#### Explanation

To solve this problem, you need to express the function $ f(x) $ as a product of linear factors. A linear factor is generally in the form $(x - a)$, where $a$ is a root of the equation.

#### Input Box and Tool Menu

In the provided image, there appears to be an empty input box next to $ f(x) = $. This is where you will input the factored form of the function.

Additionally, there is a tool menu with various mathematical symbols:

1. **i**: Likely to represent the imaginary unit.
2. **Fraction Symbol**: Indicates the option to input fractions.
3. **Power Symbol**: Allows you to input exponents.
4. **Square Root Symbol**: Permits entry of square roots.

#### Graphs or Diagrams

There are no graphs or diagrams associated with this problem statement.

#### Steps to Factor $ f(x) $

1. Identify the roots of the polynomial $ f(x) $. This could be done through techniques such as solving the equation $ f(x) = 0 $, synthetic division, or factoring by grouping.
2. Once the roots $ r_1, r_2, \dots, r_n $ are identified, express $ f(x) $ as:

\[ f(x) = a(x - r_1)(x - r_2) \cdots (x - r_n) \]

Where $ a $ is the leading coefficient of the polynomial.

#### Example

For instance, if the polynomial is $ f(x) = x^2 - 5x + 6 $, you can find that the roots are $ x = 2 $ and $ x = 3 $. Thus, the factorized form is:

\[ f(x) = (x - 2)(x - 3) \]

Remember, the exact factorization depends on the given function $ f(x) $. The provided tools can help ensure correct formatting of mathematical expressions.

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