Problem 24 - A Graphical Approach to Precalculus with Limits

Question

Find a polynomial function \(P(x)\) having leading coefficient 1, least possible degree, real coefficients. and the given zeros. \(-2+i,-2-i, 3,\) and \(-3\)

Step-by-Step Solution

Answer

The polynomial is \(P(x) = x^4 + 4x^3 - 4x^2 - 36x - 45\).

Step 1: Write Down the Zeros

The problem gives us the zeros: \(-2+i\), \(-2-i\), \(3\), and \(-3\). Complex zeros occur in conjugate pairs, so both \(-2+i\) and \(-2-i\) are needed to ensure real coefficients.

Step 2: Construct Linear Factors

Each zero \(a\) of the polynomial corresponds to a factor \((x-a)\). For the zeros \(3\) and \(-3\), the factors are \((x-3)\) and \((x+3)\). For the complex zeros \(-2+i\) and \(-2-i\), the factors are \((x - (-2+i))\) and \((x - (-2-i))\).

Step 3: Write the Polynomial

Real coefficients require us to combine complex conjugates into quadratic factors: for \(-2+i\) and \(-2-i\), form the product: \[ (x - (-2+i))(x - (-2-i)) = ((x+2) - i)((x+2) + i) = (x+2)^2 + 1 \]This simplifies to: \[(x+2)^2 + 1 = x^2 + 4x + 5\].

Step 4: Form the Complete Polynomial

Combine all factors: \[P(x) = (x-3)(x+3)(x^2 + 4x + 5)\].Each component is formed based on the zeros given.

Step 5: Simplify the Polynomial

First, multiply the first two factors: \[(x-3)(x+3) = x^2 - 9\].Now expand by multiplying with the quadratic factor:\[(x^2 - 9)(x^2 + 4x + 5) = x^4 + 4x^3 + 5x^2 - 9x^2 - 36x - 45 = x^4 + 4x^3 - 4x^2 - 36x - 45\].

Key Concepts

Complex Conjugate Zeros

In the study of polynomial functions, particularly those with real coefficients, complex zeros are quite fascinating. These zeros appear in conjugate pairs. This means that if a polynomial has a complex zero, such as \(-2+i\), then its conjugate, \(-2-i\), must also be a zero. This property is crucial because it ensures that the polynomial maintains real coefficients.

• When faced with a complex zero, always check for its conjugate.
• Complex conjugate pairs are needed to create real quadratic factors within the polynomial.

For instance, from the exercise you provided, both \(-2+i\) and \(-2-i\) are counted as zeros, leading to the formation of a quadratic factor ensuring real coefficients in the polynomial.

Real Coefficients

Real coefficients are necessary for forming polynomials that can be handled in real-world applications. When constructing a polynomial, keeping real coefficients means that all the numbers in the polynomial equation are real numbers.

• Real coefficients are achieved by ensuring complex zeros appear in conjugate pairs.
• When multiplying complex conjugate factors, the resulting coefficients will be real.

In the polynomial we constructed, combining the complex conjugate zeros creates the quadratic factor \((x^2 + 4x + 5)\), which is real. By ensuring every component is multiplied correctly, the polynomial \(P(x)\) maintains real coefficients throughout the process.

Leading Coefficient

The leading coefficient is the non-zero coefficient of the term with the highest degree in a polynomial. In many cases, like your exercise, the leading coefficient is desired to be 1 for simplicity and standardization.

• The leading coefficient is critical in determining the end behavior of a polynomial function.
• A common requirement is to have it as 1, especially when asked to present a polynomial in its simplest form.

For the given polynomial \(P(x)\), by constructing it from its zeros and keeping each step carefully executed, the leading coefficient remains 1, ensuring the polynomial is standardized as such.

Least Possible Degree

The least possible degree of a polynomial refers to the lowest degree necessary for the polynomial to have all its given zeros. The degree of a polynomial is equivalent to the number of zeros, counting multiplicity.

• Each zero of the polynomial contributes to its degree incrementally by one.
• The inclusion of complex conjugate pairs contributes two zeros, increasing the degree by 2.
• Always calculate based on the unique set of zeros.

In the exercise, the least possible degree is determined by the four zeros: \(-2+i\), \(-2-i\), \(3\), and \(-3\), leading to a polynomial of degree 4. This is the minimum degree required to have all the zeros represented in the polynomial function \(P(x)\).