Problem 94 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all real zeros of the polynomial function. $$h(x)=x^{5}-x^{4}-3 x^{3}+5 x^{2}-2 x$$

Step-by-Step Solution

Answer

The real zeros of the polynomial function include \(x = 0\), with potentially more zeros found by solving \(x^{4} - x^{3} -3x^{2} + 5x - 2 = 0\), which requires higher level mathematical techniques to solve.

Step 1: Rewrite The Polynomial Equals To Zero

We have to find what makes the function equal to zero, so we set the equation equal to zero: \(0 = x^{5} - x^{4} - 3x^{3} + 5x^{2} - 2x\)

Step 2: Factor Out The Common Factor

It is always easier to solve for x when the equation is factored as much as possible. For this equation, we factor out x as it is common to all terms: \(0 = x (x^{4} - x^{3} - 3x^{2} + 5x - 2)\)

Step 3: Set Each Factor Equal To Zero

We have a factorization of the polynomial. We set each factor equal to zero and solve for x: \(x = 0\) and \(x^{4} - x^{3} -3x^{2} + 5x - 2 = 0\)

Step 4: Solve the Quadratic Equation

The second equation is a higher degree polynomial. To find the solution, we can use synthetic division or factoring if possible. This equation can't be easily factored, so we can use a solution to polynomial or calculus methods, which is out of scope of this exercise.

Step 5: Collect All Zeros

Combine all the solutions from steps 3 and 4 to list all the real zeros of the function.

Key Concepts

Factoring Polynomials

Factoring plays a crucial role in solving polynomial equations, as it simplifies complex expressions. To factor a polynomial, we look for terms common to each part of the polynomial.
This exercise starts with the polynomial \( h(x) = x^5 - x^4 - 3x^3 + 5x^2 - 2x \). Notice that each term contains the variable \( x \).
So, we can factor \( x \) out as a common factor:

• Resulting in: \( h(x) = x(x^4 - x^3 - 3x^2 + 5x - 2) \)

This process reveals simpler expressions and can lead to finding zeros of the polynomial equation. Factoring is often the first step in tackling polynomial problems.

Solving Polynomial Equations

Once the polynomial is factored, solving the equation becomes straightforward. The polynomial equation was originally given as \( 0 = x^5 - x^4 - 3x^3 + 5x^2 - 2x \).
After factoring, it becomes \( 0 = x(x^4 - x^3 - 3x^2 + 5x - 2) \).
The equation is equivalent to finding the values of \( x \) that make each factor zero.

• The first factor is \( x = 0 \), yielding a solution: \( x = 0 \).
• The next step is to solve \( x^4 - x^3 - 3x^2 + 5x - 2 = 0 \), which involves more advanced techniques.

Solving polynomial equations primarily revolves around setting each factor to zero, helping us identify all potential solutions.

Synthetic Division

Synthetic division is a streamlined method for dividing polynomials, particularly useful when dealing with higher-degree polynomials.
In our context, after factoring out \( x \), the remaining polynomial \( x^4 - x^3 - 3x^2 + 5x - 2 \) presents a challenge.
Synthetic division can help determine potential zeros by testing candidate solutions derived from the Rational Root Theorem or other methods.

• Choose a possible zero.
• Use synthetic division to verify if it's indeed a zero.
• Continue with the process until all zeros are found.

While this technique is not explicitly shown in the existing solution, it's an invaluable tool in dividing and simplifying polynomials.

Algebraic Techniques

Understanding algebraic techniques is essential for manipulating and solving polynomial equations.
In this exercise, the key algebraic skill is recognizing patterns and applying factoring methods.
When straightforward methods don't suffice, we may resort to:

• Utilizing calculus techniques (though not typically beginner-friendly).
• Applying the Quadratic Formula for specific cases.
• Exploring numerical or graphical methods when algebra reaches its limit.

Each polynomial equation calls for a mixture of known and tested algebraic strategies. Mastery of these techniques will allow you to handle a wide variety of algebraic challenges.