images attached **Problem Statement:**Write a polynomial \( f(x) \) that satisfies the given conditions.1. Polynomial of lowest degree with zeros of \( -4 \) (multiplicity 2) and \( 1 \) (multiplicity 1).2. The function is such that \( f(0) = 96 \).**Solution:**To define the polynomial \( f(x) \), we start by expressing it in terms of its roots:- The root \( -4 \) with multiplicity 2 implies a factor of \( (x + 4)^2 \).- The root \( 1 \) with multiplicity 1 implies a factor of \( (x - 1) \).Thus, the polynomial can be expressed as:\[ f(x) = a(x + 4)^2(x - 1) \]where \( a \) is a constant to be determined.To find \( a \), we use the condition \( f(0) = 96 \):\[ f(0) = a(0 + 4)^2(0 - 1) = 96 \]\[ a \cdot 16 \cdot (-1) = 96 \]\[ a \cdot (-16) = 96 \]\[ a = \frac{96}{-16} \]\[ a = -6 \]Therefore, the polynomial is:\[ f(x) = -6(x + 4)^2(x - 1) \] **Problem Statement**Write a polynomial \( f(x) \) that satisfies the given conditions. Express the polynomial with the lowest possible leading positive integer coefficient.Polynomial of lowest degree with lowest possible integer coefficients and zeros of \( 2i \) and \( 2 - i \).**Function**\[ f(x) = \boxed{} \]**Instructions for Solving**To find the polynomial \( f(x) \):1. Note the given zeros \( 2i \) and \( 2 - i \).2. Remember that complex roots occur in conjugate pairs in polynomials with real coefficients. Thus, the zeros are \( 2i, -2i, 2 - i, \) and \( 2 + i \).3. Construct factors from these zeros: - \( (x - 2i) \) - \( (x + 2i) \) - \( (x - (2 - i)) \) - \( (x - (2 + i)) \)4. Multiply the factors to form the polynomial using the identity \( (x-a)(x+a) = x^2-a^2 \) for complex conjugates.Using these steps, you will have the polynomial expression with the lowest degree and integer coefficients.

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Answered: Write a polynomial f (x) that satisfies…

Write a polynomial f (x) that satisfies the given conditions. Polynomial of lowest degree with zeros of - 4 (multiplicity 2), 1 (multiplicity 1), and with f (0) = 96. f (x) =

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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A researcher studies water clarity at the same location in a lake on the same dates during the course of a year and repeats the measurements on the same dates 5 years later. The researcher immerses a weighted disk painted black and white and measures the depth (in inches) at which it is no longer visible. The collected data is given in the table below. Complete parts (a) through (c) below. Observation 1 2 3 4 5 6 1/25 3/19 5/30 7/3 9/13 11/7 71.2 35.6 53.8 41.2 57.3 43.4 72.1 35.1 59.0 41.0 61.0 47.0 Date Initial Depth, X, Depth Five Years Later, Y, Click the icon to view the table of critical t-values. (a) Why is it important to take the measurements on the same date? OA. Using the same dates makes it easier to remember to take samples. B. Those are the same dates that all biologists use to take water clarity samples. C. Using the same dates makes the second sample dependent on the first and reduces variability in water clarity attributable to date. D. Using the same dates maximizes…
A researcher studies water clarity at the same location in a lake on the same dates during the course of a year and repeats the measurements on the same dates 5 years later. The researcher immerses a weighted disk painted black and white and measures the depth (in inches) at which it is no longer visible. The collected data is given in the table below. Complete parts (a) through (c) below. Observation 1 2 3 4 5 6 3/19 5/30 7/3 9/13 11/77 1/25 71.2 35.6 53.8 41.2 57.3 43.4 72.1 35.1 59.0 41.0 61.0 47.0 Date Initial Depth, X, Depth Five Years Later, Y, Click the icon to view the table of critical t-values. (a) Why is it important to take the measurements on the same date? OA. Using the same dates makes it easier to remember to take samples. B. Those are the same dates that all biologists use to take water clarity samples. C. Using the same dates makes the second sample dependent on the first and reduces variability in water clarity attributable to date. D. Using the same dates maximizes…
Section 1.2 Chapter 1 0.625 0.625 -RO. I 25 0.625 0.625- 0.625 15. A manufacturer must describe a dimensioned part according to its x & y coordinates for a CNC (computer numerically controlled) machine to produce it. These coordinates are measured relative to a datum. The x- coordinate is the horizontal distance from the datum and the y-coordinate is the vertical distance from the datum. Fill in the chart that locates the holes and points that define the part. 一0.625 0.375 0.500 0.375 0.625 0.625 0.625十 1.125 0.625 1.875 0.750 RO. 1 25 I.250 l.500 O.750 DATUM 0.250 X - Coordinate Y Coordinate P5 P2 P4 H7 P6 PIO P9 H I H3 P I Pe H6 H8 H9 37
A researcher studies water clarity at the same location in a lake on the same dates during the course of a year and repeats the measurements on the same dates 5 years later. The researcher immerses a weighted disk painted black and white and measures the depth (in inches) at which it is no longer visible. The collected data is given in the table below. Complete parts (a) through (c) below. Observation 1 2 3 4 5 6 1/25 3/19 5/30 7/3 9/13 11/7 41.6 72.7 72.4 65.9 64.2 68.6 49.3 72.1 76.6 69.4 69.8 70.5 Date Initial Depth, X¡ Depth Five Years Later, Y; What is your conclusion regarding Ho? A. Do not reject Ho. There is not sufficient evidence at the α = 0.05 level of significance to conclude that the clarity of the lake is improving. B. Do not reject Ho. There is sufficient evidence at the x = 0.05 level of significance to conclude that the clarity of the lake is improving. C. Reject Ho. There is sufficient evidence at the x = 0.05 level of significance to conclude that the clarity of the…
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