Problem 98 - Algebra and Trigonometry

Question

Show that the equation $$ x^{5}-x^{4}-x^{3}-5 x^{2}-12 x-6=0 $$ has exactly one rational root, and then prove that it must have either two or four irrational roots.

Step-by-Step Solution

Answer

The rational root is \(-1\). The equation must have two or four irrational roots.

Step 1: Apply Rational Root Theorem

The Rational Root Theorem states that any rational root of a polynomial equation \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 = 0\) is a fraction \(\frac{p}{q}\) where \(p\) divides the constant term \(a_0\) and \(q\) divides the leading coefficient \(a_n\). For the polynomial \(x^5 - x^4 - x^3 - 5x^2 - 12x - 6\), the constant term is \(-6\) and the leading coefficient is \(1\). Therefore, any rational root must be a divisor of \(-6\) which are \(\pm 1, \pm 2, \pm 3, \pm 6\).

Step 2: Test Rational Roots

Testing the potential rational roots, starting with \(x = 1\): \[1^5 - 1^4 - 1^3 - 5 \times 1^2 - 12 \times 1 - 6 = 1 - 1 - 1 - 5 - 12 - 6 = -24 \] This is not zero. Next, try \(x = -1\):\[(-1)^5 - (-1)^4 - (-1)^3 - 5(-1)^2 - 12(-1) - 6 = -1 - 1 + 1 - 5 + 12 - 6 = 0\]So, \(x = -1\) is a rational root.

Step 3: Perform Polynomial Division

Since \(x = -1\) is a root, divide the polynomial by \(x + 1\) to factor it out. Using synthetic division, divide \(x^5 - x^4 - x^3 - 5x^2 - 12x - 6\) by \(x + 1\). The quotient is \(x^4 - 2x^3 - 3x^2 - 8x - 6\).

Step 4: Determine Nature of Remaining Roots

To determine the nature of the roots of the quotient \(x^4 - 2x^3 - 3x^2 - 8x - 6 = 0\), note that it has degree 4. A degree 4 polynomial can have all real irrational roots or a mix of real and non-real complex roots. Since we've factored out the only rational root, the remaining roots cannot be rational if any were present they would have been found using possible rational roots. Thus, the remaining roots are either two pairs of irrational roots or a mix of irrational and possibly complex roots.

Key Concepts

Polynomial Division

When we encounter a polynomial and find a root, particularly a rational one, we can use polynomial division to simplify the polynomial. Polynomial division works like long division for numbers.
It helps us factor out a found root to break down complex polynomials into smaller, more manageable parts.

• Consider the polynomial: \[x^5 - x^4 - x^3 - 5x^2 - 12x - 6\]
• Once a root, like \(x = -1\), is discovered, we can divide the polynomial by \(x + 1\), because \(x - (-1) = x + 1\).
• The process involves synthetic or long division, which simplifies the polynomial to a lower degree.

Performing this division correctly results in something simpler, such as \(x^4 - 2x^3 - 3x^2 - 8x - 6\), allowing further analysis on the nature of remaining roots.

Rational Roots

The Rational Root Theorem is an essential tool when dealing with polynomial equations. It helps determine if a polynomial has rational roots and what those roots might be.
According to this theorem:

• A rational root of a polynomial \(a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_0 = 0\) must be a fraction \(\frac{p}{q}\), where \(p\) is a factor of the constant term \(a_0\), and \(q\) is a factor of the leading coefficient \(a_n\).
• In our example polynomial: \(x^5 - x^4 - x^3 - 5x^2 - 12x - 6\), the constant term is \(-6\), and the leading coefficient is \(1\).
• This means possible rational roots are simple because any rational root \(\frac{p}{q}\) must be a factor of \(-6\) alone. Thus, the possible rational roots are \(\pm 1, \pm 2, \pm 3, \pm 6\).

Testing these values helps identify if they satisfy the equation, pinpointing rational roots like \(x = -1\).

Irrational Roots

After uncovering a rational root and simplifying the polynomial, we often ask about the nature of any remaining roots. If our goal is to understand the character of these roots better, it's important to consider irrational roots.
Unlike rational roots, irrational roots cannot be expressed as fractions.

• Once you've found and factored out a rational root through polynomial division, the degree of the resulting polynomial shows us how many roots remain.
• In our particular example, we've reduced to a quartic polynomial (degree 4) \(x^4 - 2x^3 - 3x^2 - 8x - 6\).
• This polynomial can have all real irrational roots or a mix of real and complex roots. But further rational roots won't exist as they've been exhausted by the theorem in the earlier step.

With a degree 4 polynomial, the most likely configurations are pairs of irrational roots. They often occur as conjugate pairs when derived from quadratic components within the polynomial's structure.