Problem 106 - Precalculus Mathematics for Calculus

Question

Give examples of polynomials that have the following properties, or explain why it is impossible to find such a polynomial. (a) A polynomial of degree 3 that has no real zeros (b) A polynomial of degree 4 that has no real zeros (c) A polynomial of degree 3 that has three real zeros, only one of which is rational (d) A polynomial of degree 4 that has four real zeros, none of which is rational What must be true about the degree of a polynomial with integer coefficients if it has no real zeros?

Step-by-Step Solution

Answer

(a) Example: \( x^3 + x + 1 \); (b) Example: \( x^4 + 4 \); (c) Example: \( (x-1)(x^2-2) \); (d) Example: \( (x^2-2)(x^2-3) \); Degree must be even for no real zeros.

Step 1: Understand the Problem Statement

The problem requires us to provide examples of polynomials meeting specific properties or explain why such polynomials cannot exist. We are tasked with finding polynomials by degree and zeros' characteristics.

Step 2: Construct Polynomial for Part (a)

For part (a), we need a cubic polynomial (degree 3) with no real zeros. Such a polynomial cannot cross the x-axis, which means its discriminant must be negative. An example of this is \( x^3 + x + 1 \). This polynomial does not have real zeros because the discriminant of this cubic polynomial is negative.

Step 3: Construct Polynomial for Part (b)

For part (b), a fourth-degree polynomial with no real zeros is required. A polynomial with entirely complex roots must have a positive discriminant and be positive for all real inputs. An example is \( x^4 + 4 \), which does not touch the x-axis because it is always positive.

Step 4: Construct Polynomial for Part (c)

For part (c), we need a cubic polynomial with three real zeros but only one rational zero. The polynomial should factor into elements where only one is rational. An example is \( (x - 1)(x^2 - 2) \), where 1 is a rational zero, and the others \( \sqrt{2} \) and \( -\sqrt{2} \) are irrational.

Step 5: Discuss Polynomial for Part (d)

In part (d), we need a degree 4 polynomial with four real zeros, none rational. One example is \( (x^2 -2)(x^2 -3) \); its roots are \( \pm \sqrt{2} \) and \( \pm \sqrt{3} \), all irrational.

Step 6: Determine Degree Condition

Polynomials with no real zeros must have an even degree and a positive discriminant, ensuring they do not intersect the x-axis. For polynomials with integer coefficients, the degree must be such that solutions are complex conjugates, typically in pairs.

Key Concepts

Degree of a Polynomial

The degree of a polynomial is a fundamental concept when working with these mathematical expressions. It refers to the highest power of the variable in the polynomial expression. For instance, in the polynomial \( x^3 + x + 1 \), the degree is 3 because the highest power of \( x \) is \( x^3 \). This degree tells us several things about the polynomial, such as:

• The number of roots, including complex ones, it can have.
• The maximum number of turning points on its graph.

Understanding the degree helps us predict how the polynomial behaves. For example, polynomials with odd degrees like 3 or 5 tend to have at least one real root, while those with even degrees can have zero real roots. This is because even-degree polynomials can start and end in the same direction, allowing them to potentially stay above or below the x-axis.

Real Zeros

Real zeros are the points where a polynomial intersects the x-axis. These are the values of \( x \) where the polynomial evaluates to zero. Real zeros provide significant insights into the polynomial's graphical representation and behavior. For example:

• A polynomial of degree 3, like \( x^3 + x + 1 \), can have up to 3 real zeros.
• Whether these real zeros are distinct or repeated affects the shape of the graph.

For a polynomial to have no real zeros, its graph must not touch or cross the x-axis at all. Such behavior is often seen in polynomials with positive discriminants, indicating that all roots are complex or in polynomials of even degree leaning towards a stable direction.

Rational and Irrational Roots

Polynomials can have roots that are either rational or irrational. Rational roots are roots that can be expressed as a simple fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers. Irrational roots, on the other hand, cannot be expressed as such fractions and often involve square roots or other non-repeating decimals.
In our problem example, part (c) asks for a polynomial of degree 3 with three real zeros, only one being rational. Consider the polynomial \( (x - 1)(x^2 - 2) \). Here:

• 1 is a rational root.
• \( \pm\sqrt{2} \) are the irrational roots.

This shows how particular structures of polynomials can give rise to a specific combination of rational and irrational roots.

Complex Roots

Complex roots occur when polynomials have no intersections with the x-axis, primarily seen in polynomials with negative discriminants. Complex roots come in conjugate pairs of the form \( a \pm bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit.
For a polynomial with degree 3, such as \( x^3 + x + 1 \), having no real roots implies all its roots are complex. A polynomial with integer coefficients must maintain its conjugate pairs when having complex roots. This ensures that the coefficients remain real.
Understanding complex roots is essential because they help define polynomial behavior even when there are no x-axis intersections. This feature applies often in control systems and signal processing, where visual representations exist but do not intersect at real coordinates.