Problem 44 - Precalculus Mathematics for Calculus

Question

Find a polynomial with integer coefficients that satisfies the given conditions. \(U\) has degree \(5,\) zeros \(\underline{4},-1,\) and \(-i,\) and leading coefficient 4 the zero \(-1\) has multiplicity 2.

Step-by-Step Solution

Answer

The polynomial is \( 4(x - 4)(x + 1)^2(x^2 + 1) \).

Step 1: Understanding the Problem

We need to find a polynomial of degree 5 with integer coefficients. It must have zeros at 4, -1 (with multiplicity 2), and -i. The leading coefficient of the polynomial must be 4.

Step 2: Identify the Zeros and Their Multiplicities

The zeros are 4, -1 (with a multiplicity of 2), and -i. From the complex conjugate root theorem, since -i is a zero, i must also be a zero. This gives us the zeros: 4, -1 (twice), -i and i.

Step 3: Write the Polynomial in Factor Form

In terms of the zeros, the polynomial can be written as:\[ U(x) = a(x - 4)(x + 1)^2(x + i)(x - i) \] where \(a\) is the leading coefficient.

Step 4: Exploit the Complex Conjugate Pairs

Notice that \((x + i)(x - i) = x^2 + 1\). This simplifies the polynomial to:\[ U(x) = a(x - 4)(x + 1)^2(x^2 + 1) \]

Step 5: Determine the Leading Coefficient

Given the leading coefficient is 4, we set \(a = 4\). Thus, the polynomial is:\[ U(x) = 4(x - 4)(x + 1)^2(x^2 + 1) \]

Step 6: Simplify the Expression

Expand the expression further to get the polynomial in standard form:1. \((x + 1)^2 = x^2 + 2x + 1\).2. Multiply \((x^2 + 1)\) and \((x^2 + 2x + 1)\).3. Multiply the result by \((x - 4)\).4. Finally, multiply the result by 4 to ensure the leading coefficient is 4.

Key Concepts

Zeros of Polynomials

Zeros of a polynomial are the values for which the polynomial evaluates to zero. In simpler terms, these are the values of the variable that make the polynomial equation equal to zero when substituted in.
Understanding zeros is crucial because they provide valuable information about the polynomial's graph and behavior. For instance, if a polynomial equation is written as \[ f(x) = (x - r_1)(x - r_2)...(x - r_n) \] each \(r_i\) (where \(i = 1, 2, ..., n\)) represents a zero of the polynomial.

• Zeros can be real or complex numbers. Real zeros are the points where the graph of the polynomial intersects the x-axis.
• If a zero has multiplicity greater than one, it indicates that the graph touches or is tangent to the x-axis instead of crossing it.
• In the original exercise, the zeros given are 4, -1 (with a multiplicity of 2), and -i. The complex conjugate root theorem helps us identify that i is also a zero.

Zeros reveal not just solutions but also hints at the polynomial's structure and symmetries. Understanding and finding them can simplify solving polynomial equations.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable in its expression. It tells us about many characteristics of the polynomial, including the maximum number of roots or zeros it can have.
The general form of a polynomial is:\[ P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \]where \(n\) is the degree of the polynomial, and \(a_n\) (the leading coefficient) is not zero.

• Polynomials of degree \(n\) can have up to \(n\) distinct zeros.
• In the exercise, the polynomial is of degree 5 because it needs to adhere to the zeros provided: 4, -1 (with multiplicity 2), and -i (& i due to complex conjugates).
• The degree indicates the number of roots (including possible repeated ones) the polynomial should have, which in this case matches the total of five zeros when considering multiplicities.

The degree also impacts the end behavior of the polynomial's graph—how it behaves as \(x\) approaches infinity or negative infinity.

Complex Conjugate Root Theorem

The complex conjugate root theorem is an essential tool when dealing with polynomials that have real coefficients. It states that if a polynomial has real coefficients and a complex number as a root, its complex conjugate is also a root.
This theorem only applies when all the coefficients are real. Suppose a polynomial given by \[ P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \] has a complex root, \( a + bi \) where \(b eq 0\). In that case, \( a - bi \) will also be a root.

• For instance, in the exercise, since \(-i\) is a zero, its conjugate \(i\) is automatically a zero because of this theorem.
• This characteristic ensures that complex roots always occur in pairs, keeping the polynomial's coefficients real when expanded.
• It simplifies the factorization of polynomials because identifying one complex root allows us to immediately include its conjugate pair.

The power of this theorem lies in its ability to streamline the process of identifying all zeros, crucial for building polynomials from known roots.