Problem 50 - Algebra and Trigonometry

Question

Find all zeros of the polynomial. \(P(x)=x^{3}-x-6\)

Step-by-Step Solution

Answer

The zeros are \(x = 2\), \(x = -1 + i\sqrt{2}\), and \(x = -1 - i\sqrt{2}\).

Step 1: Start with Rational Root Theorem

To find the zeros of the polynomial \(P(x) = x^3 - x - 6\), we can begin by applying the Rational Root Theorem. This theorem suggests that any rational root, expressed as \(\frac{p}{q}\), is such that \(p\) is a factor of the constant term \(-6\) and \(q\) is a factor of the leading coefficient \(1\). Thus, the possible rational roots are \(\pm 1, \pm 2, \pm 3, \pm 6\).

Step 2: Test Possible Rational Roots

Substitute each possible root into the polynomial \(P(x)\) to determine which, if any, are actual roots. Let's test these potential roots:- \(P(1) = 1^3 - 1 - 6 = -6\), not a root.- \(P(-1) = (-1)^3 + 1 - 6 = -6\), not a root.- \(P(2) = 2^3 - 2 - 6 = 0\), a root.- Checking others shows that none of \(-2, 3, -3, 6, -6\) are roots of the polynomial.

Step 3: Perform Polynomial Division

Using the root we found, \(x = 2\), perform synthetic division of \(P(x)\) by \((x - 2)\).The division gives us the quotient polynomial \(x^2 + 2x + 3\).

Step 4: Solve the Quotient Polynomial

Now we solve the quadratic \(x^2 + 2x + 3 = 0\) using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \(a = 1\), \(b = 2\), \(c = 3\). Substituting these values:\[x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot 3}}{2 \cdot 1} = \frac{-2 \pm \sqrt{-8}}{2}\]\[x = \frac{-2 \pm 2i\sqrt{2}}{2}\]Thus, \(x = -1 \pm i\sqrt{2}\).

Step 5: Compile All Roots

Combining all solutions from previous steps, the zeros of \(P(x) = x^3 - x - 6\) are \(x = 2\), and the complex roots \(x = -1 + i\sqrt{2}\) and \(x = -1 - i\sqrt{2}\).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a powerful tool used to identify potential rational roots of a polynomial. It helps narrow down the candidates for testing possible solutions of a polynomial equation. The theorem states that if a polynomial has rational roots, they are of the form \( \frac{p}{q} \). In this fraction, \( p \) is a factor of the constant term at the end of the polynomial and \( q \) is a factor of the leading coefficient, or the coefficient of the highest degree term. For the polynomial \( P(x) = x^3 - x - 6 \), this means:

• Constant term: \(-6\)
• Leading coefficient: \(1\)

Possible rational roots based on the theorem are \( \pm 1, \pm 2, \pm 3, \pm 6 \). Substituting each of these integers back into the polynomial helps us determine if any of them are actual roots, as a valid root will yield a result of zero when plugged into the polynomial.

Polynomial Division

Polynomial division is a method used to divide a polynomial by another polynomial, similar to the long division process with numbers. In the context of finding polynomial roots, after identifying a root using the Rational Root Theorem, polynomial division can simplify the process by reducing the degree of the polynomial.
Consider our polynomial \( P(x) = x^3 - x - 6 \) with a found root \( x = 2 \). To divide \( P(x) \) by \( (x - 2) \), synthetic division is used, which simplifies the division process. After this step, we obtain a new polynomial, called the quotient, which for this exercise is \( x^2 + 2x + 3 \). This result is a simpler quadratic equation that can be solved to find the remaining roots of the original polynomial.

• Helps simplify complex polynomials
• Reduces degree to make solving easier
• Leads to finding all polynomial roots effectively

Quadratic Formula

The quadratic formula provides an analytical method for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). It is used when factorization isn’t straightforward or when complex roots are expected. The formula is:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]For the quadratic equation derived from polynomial division, \( x^2 + 2x + 3 = 0 \), we apply the quadratic formula. Here, \( a = 1, b = 2, \) and \( c = 3 \). After substituting these into the formula, we solve for x:

• Calculate the discriminant: \( b^2 - 4ac = 4 - 12 = -8 \), indicating complex roots.
• Substitute into the formula: \( x = \frac{-2 \pm 2i\sqrt{2}}{2} \).
• Simplified roots: \( x = -1 \pm i\sqrt{2} \).

The solution reveals complex roots, which are common when the discriminant (inside the square root) is negative. These solutions complete the process of finding all roots of the polynomial.