Problem 18 - Basic Technical Mathematics with Calculus

Question

Find the remaining roots of the given equations using synthetic division, given the roots indicated. $$6 x^{4}-5 x^{3}-14 x^{2}+14 x-3=0 \quad\left(r_{1}=\frac{1}{3}, r_{2}=\frac{3}{2}\right)$$

Step-by-Step Solution

Answer

The remaining roots are \(x = \frac{-1 + \sqrt{5}}{2}\) and \(x = \frac{-1 - \sqrt{5}}{2}\).

Step 1: Understand Synthetic Division

Synthetic division is a simplified form of polynomial division, particularly useful when dividing by a linear factor of the form \(x - r\). Since we know two roots, we can use synthetic division first with \(x - \frac{1}{3}\) and then with \(x - \frac{3}{2}\).

Step 2: Prepare for Synthetic Division with First Root

Express the root \(r_1 = \frac{1}{3}\) in terms of the division factor, \(x - \frac{1}{3}\). Prepare the coefficients of the polynomial: \([6, -5, -14, 14, -3]\).

Step 3: Perform Synthetic Division for \(r_1 = \frac{1}{3}\)

Using \(x - \frac{1}{3}\), apply synthetic division to the coefficients. Multiply and add to find the reduced polynomial. Resulting coefficients are \([6, -3, -15, 9]\), meaning the quotient is \(6x^3 - 3x^2 - 15x + 9\).

Step 4: Prepare for Synthetic Division with Second Root

Take the new polynomial \(6x^3 - 3x^2 - 15x + 9\) and prepare to perform synthetic division with \(r_2 = \frac{3}{2}\), or \(x - \frac{3}{2}\). Use the new coefficients \([6, -3, -15, 9]\).

Step 5: Perform Synthetic Division for \(r_2 = \frac{3}{2}\)

Apply synthetic division using \(r_2 = \frac{3}{2}\). Work through multiplying and adding using the coefficients \([6, -3, -15, 9]\). The result will be \([6, 6, -6]\), representing \(6x^2 + 6x - 6\).

Step 6: Solve the Quadratic Equation

Simplify the polynomial \(6x^2 + 6x - 6\) by factoring out 6, resulting in \(x^2 + x - 1\). Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve \(x^2 + x - 1 = 0\).

Step 7: Calculate the Remaining Roots

Set \(a = 1\), \(b = 1\), and \(c = -1\). Calculate the discriminant: \(1^2 - 4(1)(-1) = 5\). Substitute into the quadratic formula to get \(x = \frac{-1 \pm \sqrt{5}}{2}\). Thus, the remaining roots are \(x = \frac{-1 + \sqrt{5}}{2}\) and \(x = \frac{-1 - \sqrt{5}}{2}\).

Step 8: Verify the Solution

Verify the complete set of roots as \(\frac{1}{3}, \frac{3}{2}, \frac{-1 + \sqrt{5}}{2}, \frac{-1 - \sqrt{5}}{2}\). Ensure that all roots satisfy the original polynomial equation.

Key Concepts

Synthetic Division

Synthetic division is a powerful tool for quickly and easily dividing polynomials. It's particularly useful when one of the roots of the polynomial is already known. Instead of going through the lengthy process of polynomial long division, synthetic division provides a simplified and efficient method. Imagine it as a shortcut that uses the coefficients of the polynomial and the known root to directly find the quotient and remainder.

To perform synthetic division, follow these steps:

• Write down the coefficients of the polynomial you wish to divide.
• List the known root under a divisor form, often written as \(x - r\).
• Carry out the division by performing steps of multiply and add sequentially, starting from the first coefficient.

The synthetic division process simplifies step-by-step, giving a new polynomial of one degree lower and sometimes providing a remainder of zero, which confirms that the root used is indeed a factor. This reduces the complexity of your polynomial and aids in finding additional roots or solutions.

Quadratic Formula

The quadratic formula is a tried-and-true method to find the roots of any quadratic polynomial. If you have a quadratic equation of the form \(ax^2 + bx + c = 0\), the quadratic formula can be applied to find its roots efficiently. Here's the formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Let's take a closer look at its components:

• \(a\): the coefficient of \(x^2\)
• \(b\): the coefficient of \(x\)
• \(c\): the constant term
• Discriminant: \(b^2 - 4ac\), this part under the square root sign determines the number and type of roots.

The discriminant's value can tell us a lot:

• If it is positive, the equation has two distinct real roots.
• If it is zero, there is exactly one real root.
• If negative, the roots are complex, meaning they aren't real numbers.

Utilizing the quadratic formula simplifies finding the exact roots, especially when factoring might be difficult or when dealing with irrational or complex numbers.

Polynomial Roots

Finding the roots of a polynomial involves determining the values of \(x\) for which the polynomial gives a result of zero. These roots are the solutions to the polynomial equation and are points where the graph of the polynomial crosses the x-axis.

Here’s how you can identify them effectively:

• Use Known Roots: When roots are provided, simplify the polynomial using techniques like synthetic division.
• Identify Type of Polynomial: Recognizing if a polynomial is quadratic, cubic, or otherwise can dictate which methods (e.g., factoring, quadratic formula) are suitable for finding its roots.
• Utilize the Discriminant: For quadratics, this helps predict the nature of the roots, whether they'll be real or complex.
• Solve Systematically: Use systematic approaches like polynomial division or factorization to find all roots.

The complete set of roots includes both the roots found through solving quadratic equations and any initially known roots. These can be verified to ensure that the polynomial equation is satisfied, confirming the accuracy and completeness of the solution.