Problem 31 - Precalculus with Calculus Previews

Question

Find a polynomial function \(f\) with real coefficients of the indicated degree that possesses the given zeros. degree \(5 ; 3+i,\) o (multiplicity 3)

Step-by-Step Solution

Answer

The polynomial function is \(f(x) = (x^2 - 6x + 10)x^3\).

Step 1: Understand the zeros

The function must have a degree of 5, and we are given zeros: a complex zero \(3+i\) and a zero \(o\) with a multiplicity of 3. Since polynomial functions with real coefficients require complex conjugates to also be roots, \(3-i\) must be an additional zero.

Step 2: List all zeros including multiplicity

The zeros of the polynomial are: \(3+i\), \(3-i\), \(o\) with multiplicity 3. This accounts for all 5 roots because \(3+i\) and \(3-i\) are each counted once and \(o\) is counted three times due to its multiplicity.

Step 3: Construct factors from zeros

The factors from the zeros are \((x-(3+i))(x-(3-i))\) for complex roots, and \((x-o)^3=x^3\) for the zero with multiplicity 3.

Step 4: Simplify the complex factors

Multiply out the complex roots: \((x-(3+i))(x-(3-i)) = ((x-3)-i)((x-3)+i) = (x-3)^2 - (i)^2 = x^2 - 6x + 9 + 1 = x^2 - 6x + 10\).

Step 5: Form the polynomial

Combine all factors to form the polynomial: \(f(x) = (x^2 - 6x + 10)x^3\). Multiply out the terms as necessary to ensure it reaches proper degree form.

Step 6: Verify the polynomial degree

Ensure the total degree is 5. This is confirmed as: \((x^2)(x^3) = x^5\), which matches the desired degree.

Key Concepts

Complex Zeros

In algebra, complex zeros are solutions for the polynomial equations that involve non-real numbers, specifically involving the imaginary unit, denoted as "i," where \(i^2 = -1\). Complex zeros usually appear in conjugate pairs when dealing with polynomials that have real coefficients.

• For example, if given a complex zero is \(3+i\), then \(3-i\) is also necessarily a zero of any polynomial with real numbers as coefficients.

This is crucial in the process of solving polynomial equations to ensure that the output function remains real for all real input values.
Conjugate pairs balance each other when applied to various root-finding techniques such as factoring.

Multiplicity of Roots

Multiplicity of roots in a polynomial functions refers to the number of times a particular solution, or zero, is repeated. A root's multiplicity indicates how many times that specific root contributes to the shape of the graph of the polynomial.

• If a polynomial function has a zero, \(0\), with a multiplicity of 3, it means that \((x-o)^3 = x^3\) would be a factor in the polynomial.

Higher multiplicity can make the graph of the polynomial tangent or touch the x-axis at the root without crossing it. Even multiplicities generally result in the graph simply touching the x-axis at the root, while odd multiplicities will see the graph crossing through.

Real Coefficients

Polynomials with real coefficients are equations where the coefficients of each term are real numbers. Polynomials defined in this way have crucial properties, such as the requirement to include complex conjugate roots for polynomial functions to yield real coefficients.

Real coefficients ensure that when evaluated, polynomial results produce clear geometrical interpretations in the real number system, especially when visualizing polynomial functions' graphs.

• This guarantees that every real-valued input of the function produces a real valued output, thus making the polynomial practical for most applications in real-world scenarios, including engineering and physics.

Degree of Polynomial

The degree of a polynomial is a critical feature in determining the polynomial's behavior. It is defined as the value of the highest exponent in the polynomial.

• This degree indicates the number of possible roots or zeros, considering multiplicities, that a polynomial could have.

For any polynomial of degree 5,

• we can expect to find a total of five roots (be they real, complex, or repeated), ensuring the fundamental theorem of algebra is upheld.

This concept helps in understanding both the potential intersections with the x-axis and the long-term behavior of the polynomial function in terms of end behavior and trends.