Problem 6 - Precalculus: Functions and Graphs

Question

Find a polynomial \(f(x)\) of degree 3 that has the indicated zeros and satisfies the given condition. $$-3 i, 3 i, 4 ; \quad f(-1)=50$$

Step-by-Step Solution

Answer

The polynomial is \(f(x) = -x^3 + 4x^2 - 9x + 36\).

Step 1: Identify Roots and Form the Base Polynomial

The given roots are \(-3i, 3i,\) and \(4\). Since \(-3i\) and \(3i\) are complex conjugates, they ensure the coefficients remain real. The polynomial can be written as \[ f(x) = (x + 3i)(x - 3i)(x - 4) \]The roots of the polynomial provide the factors: \(x + 3i\), \(x - 3i\), and \(x - 4\).

Step 2: Simplify the Complex Roots Product

First, simplify the product of the complex conjugate roots:\( (x + 3i)(x - 3i) = x^2 + (3i)x - (3i)x - (3i)^2 = x^2 + 9 \).Thus, the product of the complex conjugates is a real quadratic term.

Step 3: Expand to Form the Polynomial

Now, substitute the simplified quadratic back to form the polynomial:\[ f(x) = (x^2 + 9)(x - 4) \]Multiply these binomials to get:\[ f(x) = (x^2 + 9)x - (x^2 + 9)4 = x^3 + 9x - 4x^2 - 36 \]Which simplifies to:\[ f(x) = x^3 - 4x^2 + 9x - 36 \].

Step 4: Adjust Polynomial for Given Condition

Substitute \(x = -1\) into the polynomial \(f(x) = x^3 - 4x^2 + 9x - 36\) and equate it to 50:\[ (-1)^3 - 4(-1)^2 + 9(-1) - 36 = 50 \]Calculate the left side:\[ -1 - 4 - 9 - 36 = -50 \]The equation becomes \(-50 = a \times 50\). Hence, \(a = -1\).

Step 5: Finalize the Polynomial

Multiply the polynomial by \(-1\) to satisfy the condition:\[ f(x) = -1(x^3 - 4x^2 + 9x - 36) = -x^3 + 4x^2 - 9x + 36 \]. Thus, the polynomial that satisfies all the conditions is \[ f(x) = -x^3 + 4x^2 - 9x + 36 \].

Key Concepts

Complex Conjugate Roots

When dealing with polynomial functions, especially with real coefficients, complex conjugate roots become an important aspect. A complex conjugate pair consists of two roots that are mirror images of each other over the real number line.
For example, the numbers \(3i\) and \(-3i\) are complex conjugates.
Here is why they matter:

• Conjugate pairs help maintain real coefficients in the polynomial.
• Their product results in a quadratic expression with real coefficients.
• In our context, when multiplying conjugates \((x + 3i)(x - 3i)\), the result simplifies to \(x^2 + 9\), showcasing the significance of complex conjugates in simplifying polynomial equations.

These characteristics make complex conjugate roots powerful in forming polynomials that are easy to work with, maintaining both simplicity and accuracy.

Degree of a Polynomial

The degree of a polynomial is a key feature in understanding its behavior and properties. It’s defined as the highest exponent of the variable in the polynomial expression.
A polynomial of degree \(n\) means it can have up to \(n\) roots (some of which may be complex), and it can closely represent curves through \(n\) points.

• For instance, in the exercise, the polynomial degree is 3, demonstrating it can potentially have up to 3 roots.
• This degree indicates the curve's overall shape and direction.
• Higher degree polynomials are more flexible in their shape, allowing for curves, bends, and changes in direction.

Understanding the degree helps in predicting the general shape of the graph and in determining the potential number of roots, which guides efficient problem-solving in polynomial equations.

Real Coefficients

Real coefficients in a polynomial mean all the numerical factors multiplying powers of \(x\) are real numbers, as opposed to being complex. This is crucial for simplifying computations and understanding the polynomial's behavior on a real graph.

• With real coefficients, the graph of the polynomial will have symmetry properties that make it easier to visualize and analyze.
• Given that complex roots must appear in conjugate pairs, only polynomials with real coefficients ensure complex roots don't appear as standalone roots.
• In the exercise above, starting with real coefficients and maintaining them through various steps confirms consistency and readability of the polynomial.

Overall, real coefficients contribute to managing and simplifying calculations, especially in real-world scenarios where imaginary numbers may not provide tangible solutions.