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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Does this equation have two real zeroes? Why or why not?
![**Question: Analyzing the Polynomial Function**
Does the function \( f(x) = -5x^4 + 6 \) have 2 real zeroes? Why or why not?
**Explanation:**
To determine whether the function \( f(x) = -5x^4 + 6 \) has two real zeroes, we need to consider the behavior and characteristics of the polynomial.
1. **Degree and Leading Coefficient:**
- The function is a quartic polynomial (degree 4) with a leading coefficient of -5.
- A negative leading coefficient indicates the graph opens downward.
2. **Even Degree Polynomials:**
- Polynomials with an even degree and negative leading coefficients have endpoints that go to negative infinity as \( x \) approaches both positive and negative infinity.
3. **Finding the Zeroes:**
- Set \( f(x) = 0 \):
\[
-5x^4 + 6 = 0 \implies 5x^4 = 6 \implies x^4 = \frac{6}{5}
\]
- Solve for \( x \) using the fourth root:
\[
x = \pm \sqrt[4]{\frac{6}{5}}
\]
- Since there are positive and negative solutions, there are zeroes, but they are not necessarily real if considering the multiplicity in terms of sign.
4. **Number of Real Zeroes:**
- The quartic function \( x^4 = \frac{6}{5} \) results in real solutions for \( x \).
- Since the function's degree is even and it opens downward, it could cross the x-axis at most twice, depending on the roots.
Thus, the function can have 2 real zeroes, but analyzing further would involve calculating exact values and evaluating their impact on the graph. The presence of 2 or more real zeroes is corroborated by the even degree and behavior as described.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3fa757ec-9ef0-41e2-a2d2-ee98e8402c6d%2Fbe1f18c0-c921-42ae-8bda-d7c6fc441a6c%2Fy0cf3d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question: Analyzing the Polynomial Function**
Does the function \( f(x) = -5x^4 + 6 \) have 2 real zeroes? Why or why not?
**Explanation:**
To determine whether the function \( f(x) = -5x^4 + 6 \) has two real zeroes, we need to consider the behavior and characteristics of the polynomial.
1. **Degree and Leading Coefficient:**
- The function is a quartic polynomial (degree 4) with a leading coefficient of -5.
- A negative leading coefficient indicates the graph opens downward.
2. **Even Degree Polynomials:**
- Polynomials with an even degree and negative leading coefficients have endpoints that go to negative infinity as \( x \) approaches both positive and negative infinity.
3. **Finding the Zeroes:**
- Set \( f(x) = 0 \):
\[
-5x^4 + 6 = 0 \implies 5x^4 = 6 \implies x^4 = \frac{6}{5}
\]
- Solve for \( x \) using the fourth root:
\[
x = \pm \sqrt[4]{\frac{6}{5}}
\]
- Since there are positive and negative solutions, there are zeroes, but they are not necessarily real if considering the multiplicity in terms of sign.
4. **Number of Real Zeroes:**
- The quartic function \( x^4 = \frac{6}{5} \) results in real solutions for \( x \).
- Since the function's degree is even and it opens downward, it could cross the x-axis at most twice, depending on the roots.
Thus, the function can have 2 real zeroes, but analyzing further would involve calculating exact values and evaluating their impact on the graph. The presence of 2 or more real zeroes is corroborated by the even degree and behavior as described.
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