**Graphing and Factoring Polynomial Functions**The polynomial function \( f \) has zeros only at \( x = -1, 3, \) and \( 5 \). Additionally, we know that \( f(-2) \) and \( f(2) \) are positive, while \( f(4) \) and \( f(6) \) are negative.**Task A: Sketch a graph of \( f(x) \)**Below this description is a simple coordinate plane with horizontal and vertical axes labeled. The \( x \)-axis is horizontal, and the \( y \)-axis is vertical. The graph is a placeholder for the sketch of the function \( f(x) \), based on the given zeros and information about the function's positivity and negativity at certain points.**Task B: Write the equation of the polynomial \( f(x) \) in factored form.**Using the provided zeros, the polynomial \( f(x) \) can be expressed as a product of linear factors. Each zero translates to a factor of the form \( (x - \text{zero}) \). Therefore, the factored form of the polynomial is:\[ f(x) = a(x + 1)(x - 3)(x - 5) \]where \( a \) is a constant that can be determined if additional information is given.In practice, students would need to deduce further information to sketch an accurate graph and determine the value of \( a \).

we draw a graph of function f(x)

Answered: Function f has Zeros only at -1,3 and…

Function f has Zeros only at -1,3 and 5. We know that fl-2) and F(2) arepositive, white fl4) and fEC6) 'ure hegative. Sketch a gruph of Flx) b. Write the equation of polynomial flxD in factored form.

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

**Graphing and Factoring Polynomial Functions**

The polynomial function \( f \) has zeros only at \( x = -1, 3, \) and \( 5 \). Additionally, we know that \( f(-2) \) and \( f(2) \) are positive, while \( f(4) \) and \( f(6) \) are negative.

**Task A: Sketch a graph of \( f(x) \)**

Below this description is a simple coordinate plane with horizontal and vertical axes labeled. The \( x \)-axis is horizontal, and the \( y \)-axis is vertical. The graph is a placeholder for the sketch of the function \( f(x) \), based on the given zeros and information about the function's positivity and negativity at certain points.

**Task B: Write the equation of the polynomial \( f(x) \) in factored form.**

Using the provided zeros, the polynomial \( f(x) \) can be expressed as a product of linear factors. Each zero translates to a factor of the form \( (x - \text{zero}) \). Therefore, the factored form of the polynomial is:

\[ f(x) = a(x + 1)(x - 3)(x - 5) \]

where \( a \) is a constant that can be determined if additional information is given.

In practice, students would need to deduce further information to sketch an accurate graph and determine the value of \( a \).

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