Problem 37 - Algebra and Trigonometry

Question

Find a polynomial with integer coefficients that satisfies the given conditions. \(Q\) has degree 3 and zeros \(3,2 i,\) and \(-2 i\)

Step-by-Step Solution

Answer

The polynomial is \(Q(x) = x^3 - 3x^2 + 4x - 12\).

Step 1: Identify Given Zeros

The problem gives us the zeros of the polynomial Q as \(3, 2i,\) and \(-2i.\). Since complex roots come in conjugate pairs, \(2i\) and \(-2i\) are indeed conjugates, confirming they can be roots.

Step 2: Construct Linear Factors

Each zero corresponds to a linear factor of the polynomial. Thus, the zeros \(3, 2i,\) and \(-2i\) give the following factors: \((x - 3), (x - 2i),\) and \((x + 2i)\).

Step 3: Form Polynomial from Linear Factors

Multiply the linear factors together to find the polynomial: \[Q(x) = (x - 3)(x - 2i)(x + 2i)\]

Step 4: Simplify Complex Pair

First, simplify the product of the complex conjugates: \[(x - 2i)(x + 2i) = x^2 - (2i)^2 = x^2 - (-4) = x^2 + 4\]

Step 5: Combine Simplified Result with Remaining Factor

Now, the polynomial becomes: \[Q(x) = (x - 3)(x^2 + 4)\]Expand the polynomial:\[Q(x) = x(x^2 + 4) - 3(x^2 + 4)\]\[Q(x) = x^3 + 4x - 3x^2 - 12\]

Step 6: Simplify the Expression

Combine like terms to finalize the polynomial:\[Q(x) = x^3 - 3x^2 + 4x - 12\]

Key Concepts

Complex Conjugates

Complex conjugates are pairs of complex numbers. Every non-real complex number has a conjugate. The conjugate of any complex number can be found by changing the sign of the imaginary part.

• For instance, if you have the complex number \(a + bi\), its conjugate will be \(a - bi\).
• Similarly, the conjugate of \(2i\) is \(-2i\).

In polynomials with real coefficients, complex roots will always appear in conjugate pairs. This means that if a polynomial has a complex root such as \(2i\), it must also include \(-2i\) as a root. This is crucial when constructing polynomials with integer coefficients because it ensures that all terms in the polynomial remain real, maintaining the integrity of the polynomial.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable in its expression. It tells us how many solutions or roots the polynomial can have.

• A polynomial of degree 1 is called a linear polynomial.
• A polynomial of degree 2 is known as a quadratic polynomial.
• A polynomial of degree 3 is referred to as a cubic polynomial.

In our exercise, the polynomial has a degree of 3, meaning it is a cubic polynomial and can have up to three roots. This characteristic is very useful when forming or analyzing polynomials, as it provides insights into the possible structure and behavior of the polynomial.

Linear Factors

A linear factor is an expression that, when multiplied by other factors, results in a polynomial. It has the form \(x - r\), where \(r\) is one of the roots of the polynomial.

• In the exercise, the zeros are provided as 3, \(2i\), and \(-2i\).
• These zeros lead to the linear factors: \((x - 3)\), \((x - 2i)\), and \((x + 2i)\).

By multiplying these linear factors together, you rebuild the polynomial. Linear factors act as the building blocks of polynomials, and deriving them from known roots is an essential skill in algebra. They help in expressing the polynomial in a simplified and factored form that can be easily expanded to reveal its standard form.

Zeros of a Polynomial

The zeros of a polynomial, also known as roots, are the values of \(x\) that make the polynomial equal to zero. These are critical because they reveal key properties and behaviors of the polynomial.

• For example, if \(Q(x)\) is our polynomial and we substitute one of its zeros into the polynomial, it will equal zero: \(Q(3) = 0\), \(Q(2i) = 0\), \(Q(-2i) = 0\).
• In this exercise, the zeros are 3, \(2i\), and \(-2i\).

Understanding the zeros is essential for graphing a polynomial, factoring the polynomial, and solving polynomial equations. Knowing the zeros allows us to express the polynomial in terms of its roots and to uncover its behavior across various values of \(x\).