Problem 78 - College Algebra

Question

Use synthetic division to show that 5 is a solution of $$ x^{4}-4 x^{3}-9 x^{2}+16 x+20=0 $$ Then solve the polynomial equation.

Step-by-Step Solution

Answer

The solution of the given polynomial equation are the roots obtained from the original polynomial \(x^{4}-4 x^{3}-9 x^{2}+16 x+20=0\) and the roots of the reduced polynomial from the synthetic division process.

Step 1: Verify 5 as Solution Using Synthetic Division

Begin by setting up the synthetic division. Write down the coefficient of the polynomial, which are 1, -4, -9, 16, 20, and the test root, 5. The synthetic division process involves bringing the first number (1) straight down, then multiply with the test root (5), then add to the next number (-4), then again multiply the sum with test root and add to the subsequent number. Repeat it for the entire set of coefficients. The last number obtained through this process is the remainder. If it is zero, then the test root is indeed a root of the polynomial.

Step 2: Solving The Reduced Polynomial

After the synthetic division process, a reduced polynomial is obtained. The coefficients from the synthetic division form the lower-degree polynomial. In this case, the original polynomial was of the 4th degree, thus the new polynomial will be of the 3rd degree. Now, solve this polynomial equation for any remaining roots which may exist.

Step 3: Find All Roots

After finding the remaining roots from the reduced polynomial, list all roots of the original equation which includes 5 and whatever roots were obtained from the solution of the reduced polynomial.

Key Concepts

Understanding Polynomial Equations

Polynomial equations are mathematical expressions that consist of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents of variables. For instance, the equation \( x^{4}-4 x^{3}-9 x^{2}+16 x+20=0 \) is a fourth-degree polynomial equation because the highest exponent of \( x \) is 4.

Each term in a polynomial equation is called a monomial, and the degree of the polynomial is determined by the highest power of the variable present. Solving polynomial equations often means finding the values of the variable, known as roots, that make the equation true – in other words, where the polynomial evaluates to zero. These roots can be real numbers, complex numbers, or a combination of both.

Solving Polynomial Equations Effectively

To solve polynomial equations, there are several methods available, one of which is synthetic division, an efficient way to check if a given number is a root of the equation. Synthetic division simplifies the process of long division when dividing a polynomial by a binomial of the form \( x - c \).

The procedure begins with setting up the synthetic division grid using the coefficients of the polynomial and the candidate root. If, after completing the process, the remainder is zero, it implies that the candidate root is indeed a root of the polynomial. What follows is obtaining the reduced polynomial, which is the quotient of the original polynomial division. From here, you can find the remaining roots by factoring or other root-finding methods.

Discovering Roots of Polynomials

The roots of a polynomial are the solutions to the equation set to zero. They represent points where the polynomial graph touches or crosses the horizontal axis (the x-axis). In synthetic division, if the final value you obtain after the series of operations is zero, this confirms that the test root is correct. Subsequently, finding the remaining roots involves solving the lower-degree polynomial derived from synthetic division.

It's possible for a polynomial equation to have multiple roots, including repeated roots (multiplicity), and these roots could be real or complex numbers. Once all the roots are identified, they give a complete picture of the polynomial's behavior, and this information can then be used in further applications such as graphing, optimizing, and integration.