Problem 114 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. It is possible for a sixth-degree polynomial to have only one zero.

Step-by-Step Solution

Answer

False, it is not possible for a sixth-degree polynomial to have only one zero. According to the Fundamental Theorem of Algebra, a polynomial of degree n will have exactly n zeros.

Step 1: Understand the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-constant polynomial equation in one variable with complex coefficients has at least one complex root. This includes having both real and non-real roots. Furthermore, a polynomial of degree \(n\) has exactly \(n\) zeros.

Step 2: Apply this understanding to the problem

The problem is based on a sixth-degree polynomial. According to the Fundamental Theorem of Algebra, a sixth-degree polynomial will have exactly six zeros. The zeros can be either real or complex and could also be repeated.

Step 3: Conclusion

It is not possible for a sixth-degree polynomial to have only one zero because this is contrary to the Fundamental Theorem of Algebra. Therefore, the initial statement is false.

Key Concepts

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is a critical concept in mathematics that essentially has a few key points for polynomials. This theorem emphasizes that every non-constant polynomial, regardless of its degree, will have at least one complex root.
For a polynomial equation of degree \(n\), this means there are exactly \(n\) roots or zeros, considering both the visible, real roots and the invisible, non-real roots. These roots can be complex numbers, where some roots may also appear as conjugate pairs, especially if the polynomial coefficients are real numbers.
Understanding this theorem helps us determine the number of potential solutions or zeros a polynomial can have. For example, a sixth-degree polynomial must have six zeros. These zeros can encompass all real numbers, all non-real complex numbers, or a mix of both.

• Real roots signify points where the graph crosses the x-axis.
• Non-real complex roots don't have a graphical intersection with the x-axis but are numbers like \(a + bi\), where \(b eq 0\).

degree of polynomial

The degree of a polynomial is paramount in predicting the number of zeros. As defined, the degree of a polynomial is the highest power of the variable in the polynomial equation. For instance, in the polynomial \(x^6 - 3x^4 + x - 7\), the highest power is 6. Thus, it is known as a sixth-degree polynomial.
The power determines not only the shape and end behavior of the polynomial graph but also provides insights into its roots. A polynomial of degree \(n\) is expected to have exactly \(n\) roots.
These could be real or complex and may even include repeated roots. Hence, in looking at a sixth-degree polynomial, one can confidently say that there will be six zeros. It is fundamental to understand that the term "zero" refers to the roots, regardless of their nature.

• Complex roots form when the discriminant (a part of solving polynomials) is less than zero.
• Real roots can be found when the discriminant is zero or positive.

complex roots

Complex roots occur frequently when dealing with polynomials, especially under circumstances where the coefficients of the polynomial are real numbers. Real coefficients usually ensure that complex roots appear in conjugate pairs.
Conjugate pairs conjunct with each other in the format \(a + bi\) and \(a - bi\). This pairing is essential as it maintains the roots real when considered as part of the larger polynomial equation.
For a polynomial with degree \(n\), if there are complex roots, every complex root usually comes with its conjugate to balance the equation.

• Example: The roots \(2 + 3i\) and \(2 - 3i\) are a set of complex conjugate roots.
• Polynomials with real coefficients cannot have an odd number of complex roots.

Understanding complex roots involves embracing the concept of imaginary numbers. The imaginary unit, \(i\), is defined such that \(i^2 = -1\). It represents numbers that have no real location on the traditional number line but are crucial in solving polynomial equations.
Complex roots thus become an important puzzle piece in solving polynomial equations completely.