Problem 22 - College Algebra

Question

In Exercises \(17-24\) a. List all possible rational roots. b. Use synthetic division to test the possible rational roots and find an actual root. c. Use the quotient from part (b) to find the remaining roots and solve the equation. $$ 2 x^{3}-5 x^{2}-6 x+4=0 $$

Step-by-Step Solution

Answer

The roots of the equation \(2x^{3} - 5x^{2} - 6x + 4 = 0\) are \(1, 2, -1\).

Step 1: List All Possible Rational Roots

The Rational Root Theorem states the potential rational roots of a polynomial equation \(ax^n + bx^{n-1} + ...+ c = 0\) are \(\pm\) factors of c divided by factors of a. Here, a=2 and c=4. Factors of 2 are: \(\pm1, \pm2\) and factors of 4 are: \(\pm1, \pm2, \pm4\). Dividing each factor of 4 by each factor of 2 gives the list of possible rational roots: \(\pm1, \pm2, \pm4\)

Step 2: Use Synthetic Division to Find Root

Set up a synthetic division for each possible root and perform the operation. If the remainder at the end is zero, that value is a root. After testing, it is found that 1 is a root of the polynomial, because the remainder when using 1 in synthetic division is zero.

Step 3: Find Remaining Roots from Quotient of Synthetic Division

The quotient from the synthetic division performed using 1 as a root is \(2x^2 - 3x -2\). Either by factoring or using the quadratic formula \((-b \pm \sqrt{b^2 -4ac})/2a\), the roots of the quadratic equation can be found as 2 and -1. Therefore, the remaining roots of the original cubic equation are 2 and -1.

Key Concepts

Synthetic Division

Synthetic division is a simplified form of division used specifically for polynomials when dividing by a linear factor of the form \(x - k\). It's a shortcut that allows for easier calculation of the coefficients for the quotient, replacing the more cumbersome long division. To use synthetic division, follow these steps:

• Write down the coefficients of the polynomial you're dividing.
• Write the value of \(k\) to the left of the coefficients, outside of a box or a line. This \(k\) is the root you are testing for.
• Bring down the leading coefficient to the bottom row.
• Multiply this coefficient by \(k\) and write the product under the next coefficient.
• Add the values in this column and write the sum in the bottom row.
• Continue this process for all coefficients.
• The last number in the bottom row is the remainder. If it is zero, \(k\) is a root of the polynomial.

When performing synthetic division, you effectively reduce the polynomial's degree by one, which brings you closer to finding all of its roots. For instance, dividing a cubic polynomial and obtaining a quadratic quotient allows you to then apply other methods, such as factoring or the quadratic formula, to solve for the remaining roots.

Polynomial Roots

Roots of a polynomial are the values of \(x\) that satisfy the equation when the polynomial is set to zero. In other words, they are the solutions to the polynomial equation. The Rational Root Theorem is a great tool for identifying possible rational roots. It provides a list of candidates by taking the factors of the constant term and dividing them by the factors of the leading coefficient.

To find roots:

• Apply the Rational Root Theorem to list possible rational roots.
• Use methods like synthetic division or the Factor Theorem to test these potential roots.
• Once a genuine root is found, factor it out of the polynomial to reduce its degree.
• Repeat the process or use other solving methods for the remaining polynomial.

For complex polynomials, there may be real and complex roots, and their behavior is subject to the Fundamental Theorem of Algebra, which guarantees that a polynomial of degree \(n\) has exactly \(n\) roots when counted with multiplicity.

Quadratic Formula

The quadratic formula provides a direct way to find the roots of a quadratic equation of the form \(ax^2 + bx + c = 0\). It's derived from the process of completing the square. The formula \((-b \pm \sqrt{b^2 - 4ac})/(2a)\) yields the two solutions, where \(-b\) is the opposite of the \(x\) coefficient, \(-4ac\) represents four times the product of the leading coefficient and the constant term, and \(\sqrt{b^2 - 4ac}\) denotes the square root of the discriminant. The discriminant reveals the nature of the roots:

• If it's positive, there are two real and distinct roots.
• If zero, there is one real root (it's a repeated root).
• If negative, the roots are complex and come in a conjugate pair.

Using the quadratic formula is especially handy when factoring is difficult or impractical. It's a reliable method to find roots of a quadratic equation, ensuring that no solution is overlooked, which is especially useful in a process like completing the solution for a higher-degree polynomial equation, often after reducing its degree from a previous step such as synthetic division.