Problem 53 - Algebra and Trigonometry Real Mathematics, Real People

Question

A polynomial function \(f\) with real coefficients has the given degree, zeros, and solution point. Write the function (a) in completely factored form and (b) in polynomial form. Degree 3 Zeros $$-1,2+\sqrt{5} i$$ Solution Point $$f(2)=45$$

Step-by-Step Solution

Answer

The function in factored form is \(f(x) = 15(x + 1)(x - 2 - \sqrt{5}i)(x - 2 + \sqrt{5}i)\). After further steps for converting it into polynomial form (this might involve some complex number computations), the polynomial form of the function can be obtained.

Step 1: Writing the Function in Factored Form

Given that the degree of the polynomial is 3, this means there are 3 zeros (or roots). The zeros are \(-1, 2+\sqrt{5}i, 2-\sqrt{5}i\) (as complex conjugates occur as a pair). Therefore, the polynomial in factored form is \(f(x) = a(x + 1)(x - 2 - \sqrt{5}i)(x - 2 + \sqrt{5}i)\)

Step 2: Solving for the Coefficient 'a'

We know that \(f(2)=45\), substituting the x-value into the factored form results in \(a(2 + 1)(2 - 2 - \sqrt{5}i)(2 - 2 + \sqrt{5}i) = 45\). Simplifying gives \(3a(- \sqrt{5}i)(\sqrt{5}i) = 45\). Since \(a(-1)(-1) = a\), we find that \(a = 15\). So the function in factored form is \(f(x) = 15(x + 1)(x - 2 - \sqrt{5}i)(x - 2 + \sqrt{5}i)\)

Step 3: Writing the Function in Polynomial Form

To transform the factored form into polynomial form, we need to multiply the factors, simplify the expression, and group like terms together. The obtained polynomial will consist of real coefficients. This step might involve complex numbers and their properties, for example \((a+bi)(a-bi) = a^2 + b^2\). This will yield the polynomial form of the function.

Key Concepts

Complex Numbers

Complex numbers play an important role in algebra, particularly when dealing with polynomial functions. A complex number is typically written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit defined by the property \(i^2 = -1\).

These numbers allow mathematicians to work with roots of equations that have no real solutions, such as \(x^2 + 1 = 0\). The solutions to this equation are \(x = i\) and \(x = -i\). By introducing complex numbers, we can find solutions to polynomial equations that involve irrational or imaginary roots. This is essential when expressing polynomial functions in completely factored form.

In this exercise, one of the zeros of the polynomial is \(2+\sqrt{5}i\). Understanding complex numbers allows you to deal with algebraic expressions involving these seemingly abstract entries.

Complex Conjugates

Whenever a polynomial function with real coefficients has a complex zero, its complex conjugate must also be a zero. This rule ensures that when the polynomial is expanded, all coefficients remain real numbers.

A complex conjugate is formed by changing the sign of the imaginary part of a complex number. So, the complex conjugate of \(a + bi\) is \(a - bi\). This property is very useful because multiplying a complex number by its conjugate always yields a real number: \((a + bi)(a - bi) = a^2 + b^2\).

In our example, given the zero \(2 + \sqrt{5}i\), its conjugate \(2 - \sqrt{5}i\) will also be a zero of the polynomial. This ensures the polynomial maintains real coefficients, as required when expressed in factored form or polynomial form.

Polynomial Degree

Understanding the degree of a polynomial is key to determining the number of zeros it should possess. The degree of a polynomial is the highest power of the variable \(x\) in its expression when expanded.

For example, a polynomial of degree 3 can be written as \(f(x) = ax^3 + bx^2 + cx + d\), where \(a, b, c,\) and \(d\) are constants, and \(a eq 0\).

In a degree 3 polynomial, there should be exactly three zeros, which could be real or complex. These zeros could be identical or distinct. In the given exercise, the zeros are \(-1, 2+\sqrt{5}i, \) and \(2-\sqrt{5}i\). The presence of three zeros complies with the polynomial's degree and ensures the function can be expressed completely in factored form.

Factored Form

Factored form is particularly useful for identifying the zeros of a polynomial and for further algebraic simplification. In the factored form, a polynomial is expressed as the product of its factors, where each factor corresponds to a zero of the polynomial. For a polynomial function, if we know the zeros, we can write the polynomial as \(f(x) = a(x - r_1)(x - r_2)...(x - r_n)\), where \(r_1, r_2, ..., r_n\) are the zeros.

In this exercise, knowing the zeros \(-1, 2+\sqrt{5}i\), and \(2-\sqrt{5}i\), the polynomial can initially be written in factored form as \(f(x) = a(x + 1)(x - 2 - \sqrt{5}i)(x - 2 + \sqrt{5}i)\).

Factoring not only provides insight into the roots but also helps in simplifying and solving calculus-based problems, like finding derivatives and integrals. To finalize the form, finding the coefficient \(a\) by using a given solution point, like \(f(2) = 45\), helps make the polynomial expression complete.