Question

Explain why the facts given are contradictory. \(f\) is a polynomial function of degree 4 whose coefficients are real numbers; three of its zeros are \(2,1+2 i,\) and \(1-2 i\). Explain why the remaining zero must be a real number.

Short answer

The remaining zero must be a real number because the polynomial needs real coefficients and the non-real zeros already occur in conjugate pairs.

Step-by-step solution

- Understand the given information

It is given that the function f is a polynomial of degree 4 with real coefficients. The known zeros of the polynomial are 2, 1 + 2i, and 1 - 2i.

- Recall the Conjugate Root Theorem

According to the Conjugate Root Theorem, if a polynomial has real coefficients and a complex number is a zero of the polynomial, then its complex conjugate is also a zero of the polynomial.

- Identify the complex conjugate zeros

Given the zeros 1 + 2i and 1 - 2i, which are a pair of complex conjugates, we can confirm that these adhere to the Conjugate Root Theorem.

- Calculate the unknown zero

Since the polynomial is a degree 4 function, it must have exactly 4 zeros. We are already given three of them: 2, 1 + 2i, and 1 - 2i. The unknown zero must fulfill the equation having exactly 4 zeros.

- Sum up the zeros

Notice that the remaining zero of the polynomial must be real because all coefficients of the polynomial are real, and the non-real zeros already come in conjugate pairs. Consequently, there is only one remaining spot for a zero, which must ensure the polynomial retains real coefficients.

- Conclude the nature of the remaining zero

Thus, the remaining zero, in order to comply with the polynomial's real coefficients, must be a real number.

Key concepts

Zero of a Polynomial

In a polynomial function, a zero is a value of the variable that makes the polynomial equal to zero. For example, if you have a polynomial function like \( f(x) = x^2 - 4 \), the zeros are the values of \( x \) that satisfy \( f(x) = 0 \). In this case, the values are \( x = 2 \) and \( x = -2 \).
Identifying the zeros is crucial because they provide significant insight into the graph and behavior of the polynomial function. The zeros can be real or complex numbers.
When the coefficients of the polynomial are real numbers, the complex zeros must occur in conjugate pairs.

Complex Conjugate

A complex conjugate involves a pair of complex numbers that have the same real part but opposite imaginary parts. For example, the complex conjugate of \( 1 + 2i \) is \( 1 - 2i \).
This concept is particularly important in polynomial functions with real coefficients because, according to the Conjugate Root Theorem, if a polynomial has real coefficients, any non-real complex zeros must appear in conjugate pairs.
So, if \( 1 + 2i \) is a zero of the polynomial, then \( 1 - 2i \) must also be a zero to maintain real coefficients within the polynomial.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. For example, in the polynomial \( f(x) = 2x^3 + 3x^2 - x + 7 \), the degree is 3 because the highest power of \( x \) is 3.
The degree of a polynomial determines the number of zeros the polynomial will have, taking into account both real and complex zeros.
A polynomial of degree 4, for instance, will have exactly 4 zeros. In the given problem, the polynomial of degree 4 with real coefficients has three specified zeros: \( 2, 1 + 2i, \) and \( 1 - 2i \). Hence, there must be one more zero to comply with its degree.

Real Coefficients

Real coefficients in a polynomial are simply the numerical factors that multiply each of the variables, and they are real numbers. For example, in \( 3x^2 - 2x + 5 \), the coefficients are 3, -2, and 5, all of which are real numbers.
When a polynomial has real coefficients, its non-real complex zeros must appear in conjugate pairs to ensure the polynomial remains real.
This is why, in the problem statement, the complex zeros \( 1 + 2i \) and \( 1 - 2i \) are paired, and with one real zero given as \( 2 \), the remaining zero must also be real to maintain the real nature of the polynomial's coefficients.