Problem 106 - Algebra and Trigonometry

Question

How Many Real Zeros Can a Polynomial Have? 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) \(\mathrm{A}\) 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 degree 3 polynomial can have complex roots; a degree 4 with no real zeros. Degree must be even for no real zeros with integer coefficients.

Step 1: Understanding Polynomial Zeros

The zeros of a polynomial are the values of x that make the polynomial equal to zero. Real zeros are the real number solutions to this equation. A polynomial's degree gives the maximum possible number of real zeros it can have.

Step 2: Part (a) No Real Zeros for Degree 3

A polynomial of degree 3 has one or three real zeros. However, if it has one real zero, it can have complex zeros that are not real. Example: The polynomial \( x^3 + x^2 + 2 \) has no real zeros as it has one real root and two complex roots.

Step 3: Part (b) No Real Zeros for Degree 4

A polynomial of degree 4 can have no real zeros if all its roots are complex. Example: \( x^4 + 4x^2 + 5 \). The discriminant is negative, indicating no real zeros.

Step 4: Part (c) Three Real Zeros, One Rational

For a degree 3 polynomial having three real zeros, one can be rational and two can be irrational. Example: \( (x - 1)(x^2 - 2) \) has zeros 1 (rational), \( \sqrt{2} \), and \( -\sqrt{2} \) (both irrational).

Step 5: Part (d) Four Real Zeros, None Rational

A polynomial can have real irrational zeros. Example: \( (x^2 - 2)(x^2 - 3) \) equals \( x^4 - 5x^2 + 6 \) and has zeros \( \pm \sqrt{2} \) and \( \pm \sqrt{3} \), all irrational.

Step 6: Degree Condition for No Real Zeros

If a polynomial with integer coefficients has no real zeros, its degree must be even (for complex conjugate pairs without net real solutions). They need roots to pair up as complex conjugates, which requires an even degree.

Key Concepts

Real Zeros

Real zeros of a polynomial are the x-values that make the polynomial equation equal to zero, and they are actual real numbers on the number line. To find these zeros, one typically graphs the polynomial or applies algebraic methods like factoring or using the quadratic formula. Real zeros correspond to the x-intercepts of a polynomial graph.
When dealing with a polynomial, always check the degree to understand the maximum number of real zeros it can have. For example, a polynomial of degree 3, such as \( x^3 - 6x^2 + 11x - 6 \) has at most three real zeros. This doesn't guarantee all zeros are real; some might be complex.
Understanding real zeros is crucial because they reveal the behavior of the polynomial function and help in sketching its graph.

Complex Zeros

Complex zeros are solutions to polynomial equations that include imaginary numbers. They usually arise when polynomial equations have no real solutions. A polynomial's complex zeros always come in conjugate pairs; if \(a + bi\) is a zero, then \(a - bi\) is also a zero.
For instance, consider the polynomial \(x^2 + 1\). Setting this equation to zero, \(x^2 + 1 = 0\), results in roots \(x = i\) and \(x = -i\), both complex.
In polynomials with real coefficients, like \(x^3 + x^2 + 2\), complex roots appear in pairs to maintain the real nature of the polynomial's coefficients. Complex zeros are important for understanding the full spectrum of a polynomial's behaviors and solutions.

Polynomial Degree

The degree of a polynomial is the highest power of the variable in the polynomial expression. It indicates the number of solutions a polynomial equation can have, which includes both real and complex zeros.
For instance, a polynomial of degree 3 is written as \(ax^3 + bx^2 + cx + d\). It can have up to three zeros. These could be all real, all complex, or a mixture of both, depending on the polynomial's specific coefficients.
A significant indicator is that if a polynomial has integer coefficients and an odd degree, it is guaranteed to have at least one real zero. This is because polynomial graphs of odd degrees go from negative infinity to positive infinity, ensuring the graph intersects the x-axis at least once.

Integer Coefficients

Polynomials with integer coefficients are those whose coefficients (the numbers multiplying the variables) are all integers. This influences the nature of the polynomial's zeros.
If a polynomial with integer coefficients has no real zeros, it generally has an even degree. This is because the non-real roots necessarily need to occur in complex conjugate pairs (like \(a + bi\) and \(a - bi\)) for the polynomial to close up without crossing the x-axis.
Examples like \(x^4 + 4x^2 + 5\) highlight this principle; it has a degree 4, ensuring the four roots include no real numbers due to complex conjugate pairs, as derived from a negative discriminant. Understanding integer coefficients is vital for predicting and explaining the real and complex nature of zeros.