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Chapter 8: Problem 17

a. If the degree of a polynomial is odd, then at least one of its zeros must be real. Explain why this is true. b. Sketch a polynomial function that has no real zeros and whose degree is: i. 2 ii. 4 c. Sketch a polynomial function of degree 3 that has exactly: i. One real zero ii. Three real zeros d. Sketch a polynomial function of degree 4 that has exactly two real zeros.

Short Answer

Expert verified
A polynomial of odd degree must have at least one real zero due to its end behavior. A degree 2 and 4 polynomial can have no real zeros by being above or below the x-axis. A degree 3 polynomial can have one real zero or three real zeros based on its intersections. A degree 4 polynomial can have exactly two real zeros if it intersects the x-axis twice.

Step by step solution

01

Explanation for Part a

To prove why a polynomial of odd degree must have at least one real zero, consider the end behavior of the polynomial function. For large positive or negative values of x, the polynomial will either tend to positive or negative infinity. Since the polynomial is continuous, it must cross the x-axis at least once, which means it must have at least one real zero.
02

Sketch for Part b.i

To sketch a polynomial of degree 2 with no real zeros, draw a parabola that opens upwards or downwards and does not touch the x-axis. This represents a quadratic polynomial with complex roots.
03

Sketch for Part b.ii

To sketch a polynomial of degree 4 with no real zeros, draw a quartic function that opens upwards or downwards and does not touch the x-axis. This represents a polynomial with purely complex roots.
04

Sketch for Part c.i

To sketch a polynomial of degree 3 with exactly one real zero, draw a cubic function that intersects the x-axis exactly once. This demonstrates a cubic polynomial with one real root and two complex roots.
05

Sketch for Part c.ii

To sketch a polynomial of degree 3 with three real zeros, draw a cubic function that intersects the x-axis exactly three times. This shows a cubic polynomial with three distinct real roots.
06

Sketch for Part d

To sketch a polynomial of degree 4 with exactly two real zeros, draw a quartic function that intersects the x-axis at exactly two points. The function must turn twice in between the zeros to demonstrate the existence of two real roots and two complex roots.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Degree of polynomial
The degree of a polynomial is a fundamental concept. It tells you the highest power of the variable x in the polynomial. For example, in the polynomial \(3x^4 + 2x^2 + x + 5\), the degree is 4 because the highest power of x is 4. The degree gives us important information about the polynomial's behavior and properties. For instance, polynomials of degree 4 will have different characteristics compared to those of degree 3. Understanding the degree helps in predicting the number of real and complex roots, as well as how the polynomial behaves at large values of x.
Real zeros
Real zeros of a polynomial are the x-values where the polynomial crosses the x-axis. These are the solutions to the equation \(P(x) = 0\) where P(x) is the polynomial. For example, if the polynomial is \(x^2 - 4\), the real zeros are x=2 and x=-2. Finding real zeros is crucial because they are the points where the graph intercepts the x-axis.
Here are some key points about real zeros:
  • If the degree of the polynomial is odd, it must have at least one real zero. This is because the polynomial will change sign from positive to negative (or vice versa) as x goes from \(-\text{infinity}\) to \(\text{infinity}\), hence it must cross the x-axis.
  • Polynomials of even degree can have zero or more real zeros. They might just touch the x-axis or not intersect it at all.
Complex roots
Complex roots occur in conjugate pairs, meaning if \(a + bi\) is a root, then \(a - bi\) must also be a root, where i is the imaginary unit (\(i^2 = -1\)). Complex roots arise when solving polynomials that do not intersect the x-axis, indicating no real zeros. For instance, the quadratic polynomial \(x^2 + 1 = 0\) has roots \(i\) and \(-i\), which are purely complex.
Important points about complex roots include:
  • A polynomial's degree gives the total number of roots (real and complex combined).
  • When the polynomial has real coefficients, complex roots will form pairs of complex conjugates.
End behavior
The end behavior of a polynomial describes how the polynomial behaves as x approaches \(\text{infinity}\) or \(-\text{infinity}\). This is largely determined by the polynomial's leading term (the term with the highest power of x). For example, if the leading term of a polynomial is \(x^3\), as x approaches \(\text{infinity}\), \(x^3\) also approaches \(\text{infinity}\), and as x approaches \(-\text{infinity}\), \(x^3\) approaches \(-\text{infinity}\).
The rules of end behavior are:
  • For an odd-degree polynomial with a positive leading coefficient, as x approaches \(\text{infinity}\), the polynomial approaches \(\text{infinity}\), and as x approaches \(-\text{infinity}\), the polynomial approaches \(-\text{infinity}\).
  • For an even-degree polynomial with a positive leading coefficient, as x approaches both \(\text{infinity}\) and \(-\text{infinity}\), the polynomial approaches \(\text{infinity}\).
  • If the leading coefficient is negative, the end behavior is reversed.

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