Problem 45 - Precalculus : Building Concepts and Connections

Question

This set of exercises will draw on the ideas presented in this section and your general math background. One of the zeros of a certain quadratic polynomial with real coefficients is \(1+i .\) What is its other zero?

Step-by-Step Solution

Answer

The other zero of the quadratic polynomial is \(1 - i .\)

Step 1: Understanding the problem

Given that one of the zeros of a quadratic polynomial with real coefficients is \(1+i .\) A root of a polynomial is a value which when substituted into the polynomial, results in zero. Here, we are dealing with complex roots. And for quadratic polynomials with real coefficients, if one root is a complex number, then the other root is its conjugate. A conjugate of a complex number is obtained by changing the sign of its imaginary part.

Step 2: Find the conjugate

The given root is a complex number \(1+i .\) So, its conjugate will be obtained by negating the imaginary part. The conjugate is therefore \(1-i .\)

Step 3: Validate the result

The conjugate is obtained by changing the sign of the imaginary part of the given root which is \(1+i .\) The conjugate is \(1-i .\) It is known that if an equation has real coefficients and if a complex number is one root of that equation, then its conjugate will be another root. Therefore, \(1-i .\) is indeed the other zero of the quadratic polynomial.

Key Concepts

complex numbers

Complex numbers are a fascinating extension of the familiar real numbers, adding a whole new dimension to mathematical calculations. They're numbers that include a real part and an imaginary part. The standard form is written as

• \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

The imaginary unit \(i\) is defined as the square root of \(-1\). This allows for the calculation of square roots of negative numbers, which isn't possible with real numbers alone.

• For instance, solving \(x^2 + 1 = 0\) yields \(x = i\) and \(x = -i\).

Complex numbers are crucial in various fields, like engineering and physics, because they can represent quantities that oscillate, such as sound waves or electrical currents. They also ensure that all polynomial equations have solutions, a fundamental concept in algebra.

conjugates in polynomials

In the world of complex numbers, "conjugate" has a very specific meaning. For any complex number \(a + bi\), its conjugate is \(a - bi\). This is achieved by simply changing the sign of the imaginary part without touching the real part. Conjugates play a special role in polynomials, particularly those with real coefficients. When a polynomial has real coefficients, and one of its roots is complex, the other root must be the conjugate. This means:

• If \(1 + i\) is a root, \(1 - i\) must also be a root.

This property ensures that when you have complex roots, they occur in pairs. The product of these conjugates is always a real number, which satisfies the requirement of polynomials to have real solutions when their coefficients are real. Using conjugates is also helpful when simplifying expressions or solving equations involving complex numbers.

polynomial roots

The roots of a polynomial are the values that solve the polynomial equation, that is, where the polynomial evaluates to zero. For example, the roots of the quadratic polynomial \(x^2 + 2x + 1\) are the solutions to the equation \(x^2 + 2x + 1 = 0\).When dealing with quadratic polynomials, there are several possibilities for the roots:

• Both roots can be real and distinct, like in \(x^2 - 5x + 6 = 0\) with roots \(2\) and \(3\).
• The roots can be real and identical, known as repeated roots; for instance, \(x^2 - 4x + 4 = 0\) has root \(2\) twice.
• Roots can also be complex numbers. For real coefficients, these complex roots will be conjugates of each other.

Understanding roots is crucial, as they give important insights into the behavior of polynomials; they indicate where the polynomial graph will intersect the x-axis. In many practical problems, finding the roots is a key step in solving the equation.