Problem 24 - 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. $$ x^{4}-2 x^{2}-16 x-15-0 $$

Step-by-Step Solution

Answer

The roots of the given polynomial equation are \(x = -1\), \(x = 3\), and \(x = -1 \pm 2i\).

Step 1 - Identify Possible Rational Roots Using the Rational Root Theorem

The Rational Root theorem states that any possible rational root of a polynomial equation, \(p/q\), where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient. For the equation \(x^{4} - 2x^{2} - 16x - 15 = 0\), the constant term is \(-15\) and the leading coefficient is \(1\). Therefore, the possible rational roots would be \(\pm1\), \(\pm3\), \(\pm5\), and \(\pm15\).

Step 2 - Synthetic Division

The next step is to use synthetic division to find an actual root. This involves setting up a synthetic division for each potential root until a root is found whose synthetic division process results in 0 as the remainder. Here it can be found that if we choose \(x = -1\), the remainder is 0, so \(x = -1\) is a root of the polynomial. The values in the bottom row of the synthetic division represent the coefficients of the reduced polynomial \(x^{3} - x^{2} - 17x - 15 = 0\).

Step 3 - Find the Remaining Roots

From the synthetic division in step 2, we are left with the equation \(x^{3} - x^{2} - 17x - 15 = 0\). Repeat step 2 for this new equation. Using synthetic division to test our possible rational roots again, we will find \(x = 3\) leaves a remainder of 0, therefore, it is a root of this polynomial. The synthetic division will drop the degree of the polynomial to \(x^{2} + 2x + 5 = 0\). Now, solve this equation using the quadratic formula, \(x = \frac{-b\pm\sqrt{b^{2} - 4ac}}{2a}\). This gives us the remaining roots, \(x = -1 \pm 2i\).

Key Concepts

Synthetic Division

Synthetic division is a simplified method for dividing a polynomial by a linear factor. It can help determine the roots of a polynomial equation and is especially useful for finding rational roots identified using the Rational Root Theorem. To perform synthetic division, follow these steps:

• Write down the coefficients of the polynomial.
• Select a possible root and write it to the left of the coefficients.
• Bring down the first coefficient as it is.
• Multiply it by the chosen root and add it to the next coefficient.
• Continue this process until reaching the last coefficient.

When the remainder is zero, the chosen value is an actual root of the polynomial. Synthetic division not only verifies potential roots but also simplifies the polynomial for further factorization.

Polynomial Equation

A polynomial equation is an expression set to zero involving sums of powers of one or more variables multiplied by coefficients. For example, the given equation \(x^{4} - 2x^{2} - 16x - 15 = 0\)is a polynomial equation of degree four.

To solve polynomial equations:

• List all possible rational roots using the Rational Root Theorem.
• Test each possibility with synthetic division.
• Once an actual root is found, reduce the polynomial and repeat as needed.

Understanding and solving polynomial equations can reveal the behavior of various functions in algebra, calculus, and other mathematical fields.

Quadratic Formula

The quadratic formula solves quadratic equations, typically expressed in the form \(ax^2 + bx + c = 0\). After reducing a polynomial through synthetic division, if a quadratic equation remains, you can use the quadratic formula:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

• \(a\): coefficient of \(x^2\)
• \(b\): coefficient of \(x\)
• \(c\): constant term

The formula gives two roots, which can be real or complex. For instance, from the polynomial reduction in the example, the quadratic part \(x^2 + 2x + 5 = 0\) yields complex roots \(x = -1 \pm 2i\) because the discriminant \(b^2 - 4ac\) is negative, highlighting that roots can also be non-real.

Roots of a Polynomial

The roots of a polynomial are values of the variable that make the polynomial equal to zero. Finding these roots helps in understanding the solution set and behavior of the polynomial function. There can be real or complex roots, with the degree of a polynomial determining the total number of roots—including multiplicities.

To find roots:

• Apply the Rational Root Theorem to estimate possible rational roots.
• Use synthetic division to verify and locate each root.
• Once reduced to a quadratic, apply the quadratic formula to find further roots.

Real roots can be plotted on the number line, and complex roots appear as pairs. Solving for roots allows a deeper insight into the nature of polynomial equations and their graph representations.