Problem 15 - College Algebra and Calculus: An Applied Approach

Question

Find all real solutions of the polynomial equation. $$z^{4}-z^{3}-2 z-4=0$$

Step-by-Step Solution

Answer

The real solutions of the polynomial equation \(z^{4}-z^{3}-2 z-4=0\) are \(z = 0\) (multiplicity 2), \(z = 1\), and \(z = -2\).

Step 1: Factorization

Start by grouping terms: \( (z^4-z^3) - (2z+4) = 0 \). Factoring \(z^3\): from the first group and 2 from the second one yields \( z^3(z-1) - 2(z+2) = 0 \).

Step 2: Solve for z

Setting the factored equation to zero allows us to find the solutions for \(z\). The way to proceed is to set each factor equal to zero: \(z^3(z-1) = 0 \Rightarrow z = 0 \) or \( z = 1 \), and \( -2(z+2) = 0 \Rightarrow z = -2 \). So, three roots are: \(0, 1 \), and \(-2 \). These are all real numbers.

Step 3: Check missing root

Given this is a fourth degree polynomial, it should have 4 solutions. We've only found three so far. This means either one solution is missing or one of the solutions is of degree two. Further analysis shows that \(z=0\) is of degree two. We know this because the coefficient of the \(z^3\) term is zero, which indicates that \(z=0\) is a root two times over as cubic equations usually have three solutions if the coefficient of the cube term is not zero. Same applies to higher degree equations.

Key Concepts

Factorization

Polynomials can often be simplified through factorization, a process which makes it easier to find their roots. In our equation, we began by grouping terms in pairs that share common factors:

• First Group: \( z^4 - z^3 \)
• Second Group: \( -2z - 4 \)

Once terms were grouped, we factored out the highest common factors:

• From \( z^4 - z^3 \), we take out \( z^3 \), leaving \( z^3(z-1) \).
• From \( -2z - 4 \), we take out a \( -2 \), which results in \( -2(z+2) \).

The key in factorization is to reduce the polynomial into products of simpler polynomials, which can then be set equal to zero to find their roots. Each factor reveals potential solutions when solved individually.

Real Solutions

Finding real solutions of a polynomial involves determining the values of the variable that satisfy the equation. In our context, real solutions for the polynomial \( z^4 - z^3 - 2z - 4 = 0 \) are those that make the entire expression equal zero. In practical terms, here's how it is done after factorization:

• Set \( z^3(z-1) = 0 \) which gives solutions \( z = 0 \) and \( z = 1 \).
• Similarly, set \( -2(z+2) = 0 \) yielding the solution \( z = -2 \).

These solutions correspond to the real numbers where the graph of the polynomial crosses or touches the x-axis. Real solutions are vital in understanding how the polynomial behaves at specific points on a graph.

Roots of Polynomials

The roots of a polynomial are values that transform the polynomial into zero. For a polynomial of degree \( n \), there should ideally be \( n \) roots, accounting for multiplicity. Previously, we identified three roots from the equation: \( 0, 1, \) and \( -2 \).However, since we have a fourth-degree polynomial, it's expected to have four roots. This leads us to the concept of multiplicity, where a particular solution like \( z = 0 \) appears more than once.

• This particular repetition accounts for one of the missing solutions.
• Thus, \( z = 0 \) appears twice, making it a root of multiplicity 2.

Understanding the roots and their multiplicities gives deeper insight into the structure of polynomial equations and how they relate to their respective degree. Each root represents a significant point on the chart where the polynomial either crosses or merely touches the x-axis, dictated by its multiplicity.