Problem 36 - Precalculus with Calculus Previews

Question

Find a polynomial function \(f\) with real coefficients that satisfies the given conditions. degree 3 ; zeros \(1,3 i ; f(0)=27\)

Step-by-Step Solution

Answer

The polynomial function is \(f(x) = -3x^3 + 3x^2 - 27x + 27\).

Step 1: Identify the Roots

We know that the polynomial is of degree 3 and has real coefficients. The given zeros are \(1\) and \(3i\). Since the coefficients are real, the complex zeros must occur in conjugate pairs, so the zeros are \(1\), \(3i\), and \(-3i\).

Step 2: Construct the Polynomial from Zeros

The zeros of the polynomial \(1\), \(3i\), and \(-3i\) correspond to the factors \((x-1)\), \((x-3i)\), and \((x+3i)\). Therefore, the polynomial function can be expressed as \(f(x) = a(x-1)(x-3i)(x+3i)\).

Step 3: Simplify Using Complex Conjugate Pair

Simplify \((x-3i)(x+3i)\) to \((x^2 + 9)\) using the identity \((a-b)(a+b) = a^2 - b^2\). Thus, the polynomial becomes \(f(x) = a(x-1)(x^2+9)\).

Step 4: Use Given Function Value to Find Coefficient

Substitute \(x = 0\) into the polynomial \(f(x) = a(x-1)(x^2+9)\) to find \(f(0)\). This gives \(f(0) = a(0-1)(0^2+9) = a(-1)(9) = -9a\). We know \(f(0) = 27\), therefore, \(-9a = 27\). Solving for \(a\), we find \(a = -3\).

Step 5: Write the Complete Polynomial Function

Now that we have the value of \(a\), the polynomial function is \(f(x) = -3(x-1)(x^2+9)\). Expand this to find the explicit polynomial: \(f(x) = -3(x^3 + 9x - x^2 - 9) = -3x^3 + 3x^2 - 27x + 27\).

Key Concepts

Degree of a Polynomial

In mathematics, the degree of a polynomial is an important concept that tells us the highest power of the variable in the polynomial expression. It's like measuring the maximum strength of influence that the variable, often denoted as \( x \), has in the polynomial.

For example, in the polynomial \( f(x) = -3x^3 + 3x^2 - 27x + 27 \), the highest power is \( x^3 \). This makes it a polynomial of degree 3.

The degree of a polynomial gives us valuable information, such as:

• The number of roots or zeros it can have. A degree 3 polynomial can have up to 3 roots.
• The behavior of the function as \( x \) goes to infinity. Higher degrees often mean steeper curves.
• The shape of the graph, which includes the number of turns it can take.

Understanding the degree helps us see the broad scope of possible behaviors for the function.

Complex Conjugates

Complex conjugates come into play when dealing with polynomials that have complex roots, especially when the polynomial has real coefficients.

A complex number is typically represented as \( a + bi \), where \( i \) is the imaginary unit (\( i^2 = -1 \)). Its complex conjugate is \( a - bi \). When a polynomial with real coefficients has a complex root \( 3i \), its complex conjugate \( -3i \) must also be a root. This ensures that the polynomial remains real.

Consider the given polynomial problem where the zeros are \( 1 \), \( 3i \), and by the rule of complex conjugates, also \( -3i \). These pairs ensure that when we multiply out the factors \( (x-3i)(x+3i) \), the imaginary components will cancel each other out, resulting in a real-valued polynomial. This cancellation results in \( x^2 + 9 \), maintaining real coefficients.

Polynomial Roots

A polynomial root, also known as a zero, is a value for which the polynomial equals zero. These roots are crucial as they tell us where the polynomial crosses the x-axis on a graph.

For the exercise problem, the roots provided were \( 1 \), \( 3i \), and \( -3i \). Each root corresponds to a factor in the polynomial function expression. For instance, the root \( 1 \) gives us the factor \( (x-1) \).

There are several key points about polynomial roots:

• A degree 3 polynomial has three roots, which could be a mix of real and complex numbers.
• Repetitions of roots, known as multiplicities, affect the shape and nature of the polynomial’s graph.
• The sum of the roots is related to the coefficients of the polynomial, due to Vieta's formulas.

Being aware of the roots and their multiplicity helps predict the function’s behavior.

Real Coefficients

In polynomial functions, real coefficients ensure that the terms of the polynomial have real numbers. This is vital for the properties of the polynomial, especially in terms of roots and graphing.

Real coefficients dictate that if there are any complex roots, they must come in conjugate pairs. This was seen in the exercise where the roots were \( 1 \), \( 3i \), and \( -3i \). The presence of \( 3i \) mandated a conjugate root \( -3i \) to keep the polynomial function with real coefficients.

Here's why real coefficients are important:

• They simplify graphing the polynomial as the entire graph is defined in the real plane.
• Real values result in polynomials that naturally align with real-world data, making them practical for modeling real situations.
• They ensure that outputs for real inputs are also real, an essential property in many applications including physics and engineering.

Overall, real coefficients create a balance, allowing complex roots to cancel out imaginary components and keep the solution straightforward and applicable.