Question

Explain why the facts given are contradictory. For the polynomial function \(f(x)=x^{2}+2 i x-10:\) (a) Verify that \(3-i\) is a zero of \(f\). (b) Verify that \(3+i\) is not a zero of \(f\). (c) Explain why these results do not contradict the Conjugate Pairs Theorem.

Short answer

The function \(f(x)\) has complex coefficients, so the Conjugate Pairs Theorem does not apply.

Step-by-step solution

Verify that 3-i is a zero of f(x)

Substitute \(x = 3-i\) into \(f(x) = x^2 + 2ix - 10\). Then, check if the result equals zero. \(f(3-i) = (3-i)^2 + 2i(3-i) - 10\) Expand and simplify:\(f(3-i) = (9 - 6i + i^2) + (6i - 2i^2) - 10\)\(f(3-i) = 9 - 6i - 1 + 6i + 2 - 10\)\(f(3-i) = 0\)Thus, \(3-i\) is a zero of \(f(x)\).

Verify that 3+i is not a zero of f(x)

Substitute \(x = 3+i\) into \(f(x) = x^2 + 2ix - 10\). Then, check if the result equals zero. \(f(3+i) = (3+i)^2 + 2i(3+i) - 10\) Expand and simplify:\(f(3+i) = (9 + 6i + i^2) + (6i + 2i^2) - 10\)\(f(3+i) = 9 + 6i - 1 + 6i - 2 - 10\)\(f(3+i) = 8 + 12i - 12\)\(f(3+i) eq 0\)Thus, \(3+i\) is not a zero of \(f(x)\).

Explain why there is no contradiction with the Conjugate Pairs Theorem

The Conjugate Pairs Theorem applies to polynomials with real coefficients, stating that if a polynomial has real coefficients and a complex root \(a + bi\), then its complex conjugate \(a - bi\) must also be a root. In this case, the polynomial \(f(x) = x^2 + 2ix - 10\) has complex coefficients, specifically the term \(2ix\). Therefore, the Conjugate Pairs Theorem is not applicable, and there is no contradiction.

Key concepts

Conjugate Pairs Theorem

The Conjugate Pairs Theorem is an essential concept in polynomial functions, particularly when dealing with complex numbers as roots. It states that if a polynomial function has real coefficients and possesses a complex root, then the complex conjugate of that root must also be a root of the polynomial. A complex number in the form of \(a + bi\) has its complex conjugate as \(a - bi\).

For example:
If \((3 - i)\) is a zero of a polynomial function with real coefficients, then \((3 + i)\) must also be a zero. Conversely, if a polynomial function does not have real coefficients, the theorem does not apply.

This theorem is particularly useful in narrowing down potential roots when solving polynomial equations, as these roots often come in complex conjugate pairs. However, the theorem strictly requires the polynomial to have real coefficients.

Complex Numbers

Complex numbers expand our understanding beyond the real number line by introducing an imaginary component. A complex number is written in the form \(a + bi\), where:

• \(a\) is the real part.
• \(b\) is the imaginary part.
• \(i\) is the imaginary unit, satisfying \(i^2 = -1\).

Complex numbers allow for solutions to certain equations that do not have real solutions, such as \(x^2 + 1 = 0\). The roots of this equation are \(i\) and \(-i\).

They follow specific arithmetic rules for addition, subtraction, multiplication, and division. When calculating with complex numbers, it's vital to correctly manage the imaginary unit \(i\).
For example:
\((3 - i)^2\) can be expanded as: \( (3 - i)(3 - i) = 9 - 6i + i^2 = 9 - 6i - 1 = 8 - 6i \). Proper manipulation of these can lead to recognizing complex roots in polynomials.

Polynomial Functions

Polynomial functions are mathematical expressions that consist of variables raised to non-negative integer powers, with coefficients that can be real or complex. A general form of a polynomial in one variable is:

\[ P(x) = a_nx^n + a_{n-1}x^{n-1} + \fast a_1x + a_0 \]

The degree of the polynomial is determined by the largest exponent \(n\). Coefficients \(a_n, a_{n-1}, \fast a_0\) can be real or complex numbers.

Polynomial functions exhibit a variety of properties:

• Continuity: They are continuous over the real number line.
• Derivatives: Polynomials have derivatives of all orders.
• Roots: They can have real or complex roots.

In the given exercise, the polynomial \(f(x) = x^2 + 2ix - 10\) has complex coefficients. This distinction is crucial because the Conjugate Pairs Theorem does not apply to polynomials with complex coefficients. As a result, the contradiction observed in the verification steps is not valid under this theorem. This insight helps clarify how polynomial coefficients impact their roots and the applicability of certain mathematical theorems.