Problem 104 - Algebra 2

Question

A polynomial equation with integer coefficients has the given roots. What additional roots can you identify? \(-i, 4 i\)

Step-by-Step Solution

Answer

The additional roots are i and -4i.

Step 1: Identify the Conjugate Roots

Given roots are -i and 4i. The conjugate of -i is i and the conjugate of 4i is -4i. So, these are the additional roots that can be identified.

Step 2: Write the equation in standard form

Rearrange and simplify the equation.

Step 3: Apply the solution method

Use factoring, quadratic formula, substitution, or other methods.

Step 4: Verify the solution(s)

Check solutions in the original equation.

Step 5: State the final answer

List all valid solutions.

Step 6: Conclude with the answer

The additional roots are i and -4i.

Key Concepts

Integer Coefficients

Integer coefficients are fundamental when working with polynomials. A polynomial has integer coefficients if all the numbers in front of the variables (like 3 in the term 3x) are whole numbers without fractions or decimals. These numbers can be positive, negative, or zero.

When dealing with polynomials with integer coefficients, any roots that are non-real, such as imaginary or complex numbers, must occur in conjugate pairs. This is because when you multiply conjugate pairs, the imaginary parts cancel out, yielding a polynomial with real (and potentially integer) coefficients.

Imagine a polynomial with roots that include the imaginary number \(-i\). To keep the polynomial's coefficients as integers, \(i\) would also need to be a root, so that their product is a real number. This rule is very helpful in ensuring that the polynomial maintains the necessary structure and characteristics.

Complex Conjugate Roots

Complex conjugate roots are pairs of complex numbers that have identical real parts and opposite imaginary parts. For a complex number, a + bi, its conjugate is a - bi.

Polynomials with real or integer coefficients require these roots to occur in conjugate pairs. For example:

• The complex root 4i must be partnered with the root -4i.
• Similarly, the root -i will have its pair as i.

This ensures that any imaginary components balance out, maintaining a polynomial with real (or integer) coefficients. This property is crucial as it enables mathematicians to deal with polynomials systematically.

By understanding this concept, you can predict or derive additional roots of a polynomial equation, maintaining essential mathematical properties.

Imaginary Numbers

Imaginary numbers are numbers that can be written as a real number multiplied by the imaginary unit i, where \(i\) is defined as the square root of -1, \(i^2 = -1\). Imaginary numbers are used to expand our understanding of number systems, allowing us to solve equations that do not have real roots.

When imaginary numbers come into play in polynomial equations, they often appear as part of complex numbers, which have the form a + bi, where both a and b are real numbers. Here, i is the imaginary part. This was introduced to manage square roots of negative numbers within the realm of mathematical operations.

Despite sounding complex, these numbers are incredibly useful in a variety of fields ranging from engineering to physics. They allow for solutions to polynomial equations that would otherwise be unsolvable, broadening the horizon of mathematical and practical problem-solving. For instance, in the case of our given roots, \(-i\) and \(4i\), we use their properties to confirm the polynomial's nature and predict additional roots using conjugate relationships.