Problem 98 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all real zeros of the polynomial function. $$h(x)=x^{5}+5 x^{4}-5 x^{3}-15 x^{2}-6 x$$

Step-by-Step Solution

Answer

One real root for the original equation is \(x = 0\), and the others can be found by solving the higher degree polynomial \(x^{4} + 5x^{3} - 5x^{2} - 15x - 6\) equation.

Step 1: First arrange the polynomial

We have the equation \(h(x) = x^{5} + 5x^{4} - 5x^{3} - 15x^{2} - 6x\). This can be rearranged to form \(x^{5} + 5x^{4} - 5x^{3} - 15x^{2} - 6x = 0\).

Step 2: Identify common factors

Notice that in this equation, all terms have \(x\) as a common factor. Factor \(x\) out from each term. The equation becomes \(x (x^{4} + 5x^{3} - 5x^{2} - 15x - 6) = 0\).

Step 3: Solve for x

Since the product of two numbers equals zero only if at least one of the numbers is zero, set \(x = 0\) and \(x^{4} + 5x^{3} - 5x^{2} - 15x - 6 = 0\).

Step 4: Finding roots of the second equation

This is a higher order polynomial equation where \(x = 0\) is one of the roots. The other roots can be found using methods such as synthetic division or Rational Root Theorem. Which method to use generally depends on the given situation. In this case, trial and error can also be used as it is not a third-degree equation.

Key Concepts

Factoring Polynomials

Factoring polynomials is a vital technique in algebra that helps simplify expressions and solve polynomial equations. At its core, factoring involves expressing a polynomial as a product of its smaller components, or factors. In the exercise you provided, we started with the polynomial \[ h(x) = x^{5} + 5x^{4} - 5x^{3} - 15x^{2} - 6x \] and noticed that each term had an 'x' in common. By factoring out this common term, the polynomial becomes more manageable:\[ h(x) = x (x^{4} + 5x^{3} - 5x^{2} - 15x - 6) \] Once factored, solving for zeros becomes simpler, as you can set each factor to zero. This gives the first zero directly from the factor 'x' in this case, which is \[ x = 0 \] For the remaining expression, further factoring or other techniques may be necessary to find all zeros.

Rational Root Theorem

The Rational Root Theorem is a useful tool in finding potential rational solutions to polynomial equations. It states that any rational root of a polynomial equation with integer coefficients will be a fraction \(\frac{p}{q}\) where:

• \(p\) is a factor of the constant term
• \(q\) is a factor of the leading coefficient

Applying this theorem involves listing all possible values of \(\frac{p}{q}\) and testing each one to determine if it is a root. In the context of our polynomial \[ x^{4} + 5x^{3} - 5x^{2} - 15x - 6 = 0 \] The Rational Root Theorem helps identify potential rational zeros that, when plugged into the equation, should yield zero. Testing these values can be done quickly via direct substitution or synthetic division, helping to isolate the true zeros from the potential candidates.

Higher Order Polynomials

Higher order polynomials are polynomials of degree three or higher. Solving these can often be more complex than quadratic equations. For instance, solving a fourth degree polynomial or quartic, such as \[ x^{4} + 5x^{3} - 5x^{2} - 15x - 6 \] may require various approaches. Some of the most common techniques include:

• Factoring: If a polynomial can be factored, it simplifies finding roots.
• Graphical Analysis: Plotting can give hints on the number and approximate locations of the roots.
• Numerical Methods: Techniques like synthetic division or trial and error to test for real zeros.

Each of these methods has its use depending on the specific problem. For the exercise at hand, we observed that known techniques like the Rational Root Theorem and factoring by common terms were instrumental in narrowing down and identifying zeros efficiently.