Problem 35 - Precalculus with Calculus Previews

Question

Find a polynomial function \(f\) with real coefficients that satisfies the given conditions. degree \(2 ;\) zero \(1+i ; f(1)=5\)

Step-by-Step Solution

Answer

The polynomial function is \(f(x) = 5x^2 - 10x + 10\).

Step 1: Identify the Zeros

Since polynomial functions with real coefficients have complex roots in conjugate pairs, and given that the zero is \(1+i\), we know the other zero must be \(1-i\). Thus, the zeros of the polynomial are \(x_1 = 1+i\) and \(x_2 = 1-i\).

Step 2: Construct the Polynomial from Zeros

The polynomial function \(f(x)\) can be expressed using its zeros. Since both zeros are \(x_1 = 1+i\) and \(x_2 = 1-i\), we construct \(f(x) = (x - (1+i))(x - (1-i))\).

Step 3: Expand the Expression

Expand the expression for \(f(x)\):\[(x - (1+i))(x - (1-i)) = ((x-1)-i)((x-1)+i)\]Use the difference of squares formula:\[ ((x-1)-i)((x-1)+i) = (x-1)^2 - (i)^2\]Since \(i^2 = -1\), it becomes:\[ (x-1)^2 + 1\]

Step 4: Simplify the Polynomial

Now we simplify \((x-1)^2 + 1\):\[(x-1)^2 = x^2 - 2x + 1\]So, the polynomial is:\[f(x) = x^2 - 2x + 1 + 1 = x^2 - 2x + 2\]

Step 5: Determine the Leading Coefficient

Given \(f(1) = 5\), let's determine the leading coefficient, \(k\), to satisfy this condition. So we write \(f(x) = k(x^2 - 2x + 2)\) and solve for \(k\) using \(f(1) = 5\):\[f(1) = k((1)^2 - 2(1) + 2) = 5\]\[f(1) = k(1 - 2 + 2)\]\[f(1) = k(1) = 5 \\Rightarrow k = 5\]

Step 6: Write the Final Polynomial

Substitute \(k=5\) back into the polynomial:\[ \f(x) = 5(x^2 - 2x + 2) \]\Simplified, this becomes:\[ f(x) = 5x^2 - 10x + 10 \]

Key Concepts

Complex Conjugates

In the world of polynomial functions, complex numbers play an important role, especially when the polynomials have real coefficients. Complex numbers are formed by combining real numbers and imaginary numbers, where the imaginary number is represented by the letter \(i\), satisfying \(i^2 = -1\).

When dealing with polynomials with real coefficients, complex roots always come in pairs called complex conjugates. A complex conjugate of a number is created by changing the sign of the imaginary part of the number.
For example, if we are given one root as \(1+i\), its complex conjugate pair will be \(1-i\). This is because the imaginary part, \(+i\), becomes \(-i\) for the conjugate.

Here are some essential points about complex conjugates:

• They ensure when they multiply together and result in a real number.
• Using conjugates helps in easily forming polynomials with real coefficients.
• For polynomials, complex roots appear as conjugates for the function to then have real coefficients.

Understanding complex conjugates is crucial for solving polynomial equations, ensuring that all roots of the function are accounted for, especially when the polynomial has real coefficients.

Zeros of Polynomials

Zeros of polynomials are the values of \(x\) for which the polynomial evaluates to zero. These values are also often called roots or solutions of the polynomial equation.

When constructing polynomial functions, knowing the zeros is essential as they determine the key characteristics of the polynomial function such as its shape and behavior on a graph.

The given exercise mentions zeros as \(1+i\) and \(1-i\). Here's how zeros influence polynomial construction:

• Each zero \(x_i\) from a polynomial provides a factor \((x-x_i)\).
• If a polynomial is of degree \(n\), it will have \(n\) zeros or roots. In our case, the polynomial is degree 2, so there are two zeros specified.
• Multiplying these factors provides the foundational structure from which a polynomial can be expanded.

This understanding is integral in forming a polynomial from its roots. Specifically, the knowledge from zeros helps us to write the polynomial as \((x-(1+i))(x-(1-i))\) for our exercise, giving us a structured form to expand and build the polynomial.

Degree of Polynomial

The degree of a polynomial is one of its most fundamental characteristics. It is defined as the highest power of \(x\) within the function, and it provides crucial insight into the number and types of roots the polynomial will possess.

For the exercise, it's specified that the polynomial we deal with has a degree of 2. Here’s why this is important:

• The degree tells us the maximum number of zeros or roots the polynomial will have, which can be real or complex.
• In a quadratic polynomial (degree 2), the graph is a parabola, which can open upwards or downwards based on the leading coefficient.
• A polynomial degree indicates the number of turning points or changes in direction it can have on a graph.

This degree not only helps to set the expectation of roots but also influences the general form and complexity of the polynomial equation. Understanding the degree is valuable for anticipating the characteristics of both the polynomial as an equation and its representation on a graph.