Homework Help is Here – Start Your Trial Now!
Math
Algebra
A polynomial P(x) has degree 3 and leading coefficient -6. Two of its roots are -42 and 1. Find a possible formula for P(x). You do not need to simplify/distribute your answer. P(x) =

A polynomial P(x) has degree 3 and leading coefficient -6. Two of its roots are -42 and 1. Find a possible formula for P(x). You do not need to simplify/distribute your answer. P(x) =

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section: Chapter Questions
Problem 28RE
See similar textbooks
icon
Related questions
Question
**Problem Statement:** *A polynomial \( P(x) \) has degree 3 and leading coefficient \(-6\). Two of its roots are \(-4i\) and 1.* *Find a possible formula for \( P(x) \). You do not need to simplify/distribute your answer.* **Solution:** To find the polynomial \( P(x) \), we start with the given roots and leading coefficient. Since \( P(x) \) is a polynomial of degree 3, it must have three roots. The given roots are \(-4i\) and 1. For complex roots, their conjugates also must be roots of the polynomial if the coefficients are real. Hence, the conjugate of \(-4i\) is \(4i\). So, the roots of the polynomial \( P(x) \) are: 1. \(1\) 2. \(-4i\) 3. \(4i\) Using these roots and the fact that the leading coefficient is \(-6\), we can construct the polynomial. If a polynomial with roots \( a, b, \) and \( c \) is factored, it looks like this \( k(x - a)(x - b)(x - c) \), where \( k \) is the leading coefficient. Therefore, a possible polynomial \( P(x) \) can be written as: \[ P(x) = -6 (x - 1) \left(x - (-4i)\right) (x - 4i) \] This yields: \[ P(x) = -6 (x - 1) (x + 4i) (x - 4i) \] Using the difference of squares on the imaginary numbers: \[ (x + 4i)(x - 4i) = x^2 - (4i)^2 = x^2 + 16 \] Thus, the polynomial can be written as: \[ P(x) = -6 (x - 1)(x^2 + 16) \] This is the factored form of the polynomial \( P(x) \) with the given properties.
Transcribed Image Text:**Problem Statement:** *A polynomial \( P(x) \) has degree 3 and leading coefficient \(-6\). Two of its roots are \(-4i\) and 1.* *Find a possible formula for \( P(x) \). You do not need to simplify/distribute your answer.* **Solution:** To find the polynomial \( P(x) \), we start with the given roots and leading coefficient. Since \( P(x) \) is a polynomial of degree 3, it must have three roots. The given roots are \(-4i\) and 1. For complex roots, their conjugates also must be roots of the polynomial if the coefficients are real. Hence, the conjugate of \(-4i\) is \(4i\). So, the roots of the polynomial \( P(x) \) are: 1. \(1\) 2. \(-4i\) 3. \(4i\) Using these roots and the fact that the leading coefficient is \(-6\), we can construct the polynomial. If a polynomial with roots \( a, b, \) and \( c \) is factored, it looks like this \( k(x - a)(x - b)(x - c) \), where \( k \) is the leading coefficient. Therefore, a possible polynomial \( P(x) \) can be written as: \[ P(x) = -6 (x - 1) \left(x - (-4i)\right) (x - 4i) \] This yields: \[ P(x) = -6 (x - 1) (x + 4i) (x - 4i) \] Using the difference of squares on the imaginary numbers: \[ (x + 4i)(x - 4i) = x^2 - (4i)^2 = x^2 + 16 \] Thus, the polynomial can be written as: \[ P(x) = -6 (x - 1)(x^2 + 16) \] This is the factored form of the polynomial \( P(x) \) with the given properties.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 4 steps with 4 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra: Structure And Method, Book 1
Algebra: Structure And Method, Book 1
Algebra
ISBN:
9780395977224
Author:
Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:
McDougal Littell
Calculus For The Life Sciences
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS