Problem 30 - A Graphical Approach to College Algebra

Question

Find a polynomial function \(P(x)\) having leading coefficient \(1,\) least possible degree, real coefficients, and the given zeros. $$5 (\text { multiplicity } 2 )\text { and } -2 i$$

Step-by-Step Solution

Answer

The polynomial is \(P(x) = x^4 - 10x^3 + 29x^2 - 100x + 100\).

Step 1: Write down the given zeros and their properties.

We have two given zeros: \(5\) with a multiplicity of 2 and \(-2i\), which is non-real.

Step 2: Identify Necessary Conjugate Pairs

Since \(-2i\) is a non-real root, its complex conjugate, \(2i\), must also be a root for the polynomial to have real coefficients.

Step 3: Express each root as a factor

The factor corresponding to \(5\) with multiplicity 2 is \((x - 5)^2\). For the roots \(-2i\) and \(2i\), the factors are \((x + 2i)\) and \((x - 2i)\).

Step 4: Simplify the product of complex conjugate factors

Multiply \((x + 2i)\) and \((x - 2i)\) to get a real factor: \[(x + 2i)(x - 2i) = x^2 - (2i)^2 = x^2 + 4\]

Step 5: Construct the polynomial

Combine all factors to form the polynomial: \[P(x) = (x - 5)^2(x^2 + 4)\].

Step 6: Expansion and simplification

First expand \((x - 5)^2\):\[(x - 5)^2 = x^2 - 10x + 25\].Then the polynomial:\[P(x) = (x^2 - 10x + 25)(x^2 + 4)\].Multiply these polynomials and simplify to:\[P(x) = x^4 - 10x^3 + 29x^2 - 100x + 100\].

Key Concepts

Complex Conjugates

Complex numbers consist of a real and an imaginary part. When dealing with polynomials with real coefficients, complex roots must appear in conjugate pairs. A complex conjugate pair is composed of numbers like \(a + bi\) and \(a - bi\), where \i\ is the imaginary unit. Having complex conjugate pairs ensures that when the polynomial is expanded, all terms involving \i\ cancel out. This avoids imaginary numbers in the polynomial's coefficients, keeping them real.

In the given problem, because \text{-2i}\ is a root, its complex conjugate \(2i\) must also be a root. This balancing act allows us to use complex roots without compromising the requirement for real coefficients in the polynomial.

Zeros and Multiplicity

In polynomial functions, zeros are values where the polynomial equals zero. These values are also known as roots or solutions. Multiplicity refers to the number of times a particular zero appears.

For instance, if a zero has a multiplicity of 2, like the zero \(5\) in our exercise, it means \(5\) is a root twice. Geometrically, this implies the polynomial will just touch, but not cross, the x-axis at \(x = 5\).

This concept is crucial because the multiplicity affects the shape and degree of the polynomial. If not given directly, multiplicity can often be inferred from factors of the polynomial.

Real Coefficients

Polynomial coefficients are the numbers that multiply the variable terms. Real coefficients denote that every coefficient in the polynomial is a real number, as opposed to an imaginary one.

For a polynomial to have real coefficients, if any of its roots are complex, these roots must appear in complex conjugate pairs. This ensures that any imaginary parts cancel out once the polynomial is expanded, leaving behind only real numbers as coefficients.

This requirement is especially important in real-world applications where mixing real and complex numbers in coefficients would not be meaningful or applicable.

Polynomial Degree

The degree of a polynomial is the highest power of the variable present in the polynomial equation. It gives an idea of the polynomial's complexity and graph shape. The polynomial's degree directly correlates with the total number of its roots if you count each according to its multiplicity.

For instance, in the polynomial \(P(x) = x^4 - 10x^3 + 29x^2 - 100x + 100\), the degree is 4. This can be deduced from the highest power of \(x\) in the polynomial. This degree reflects on the possible number of turning points and the behavior of the polynomial as \(x\) approaches infinity.

Understanding the polynomial degree helps in sketching the graph of the polynomial and predicting its behavior, making it easier to solve and interpret real-world problems.