Problem 46 - College Algebra and Calculus: An Applied Approach

Question

Use synthetic division to find each function value. \(g(x)=x^{6}-4 x^{4}+3 x^{2}+2\) (a) \(g(2)\) (b) \(g(-4)\) (c) \(g(7)\) (d) \(g(-1)\)

Step-by-Step Solution

Answer

The function values are: (a) \(g(2) = 28\), (b) \(g(-4) = -702\), (c) \(g(7) = 5845\), and (d) \(g(-1) = 2\).

Step 1: Synthetic Division for g(2)

In synthetic division, the number 2 is written to the left and then the coefficients of the polynomial are written in a row. In the first step 2 is brought down, then it is multiplied by 2 (the number to the left) and the result is written in the second row. This process is repeated for each coefficient to arrive at the sum which is the value of the polynomial at 2. Here is the table for the synthetic division: \[\begin{array}{c|ccccc} 2 & 1 & 0 & -4 & 0 & 3 & 2 \ & & 2 & 4 & -8 & -16 & 26 \ \hline & 1 & 2 & 0 & -8 & -13 & 28 \end{array}\] Thus, the value of \(g(2)\) is 28.

Step 2: Synthetic Division for g(-4)

The process is repeated with -4 to find \(g(-4)\): \[\begin{array}{c|ccccc} -4 & 1 & 0 & -4 & 0 & 3 & 2 \ & & -4 & 16 & -48 & 180 & -704 \ \hline & 1 & -4 & 12 & -48 & 183 & -702 \end{array}\] Thus, the value of \(g(-4)\) is -702.

Step 3: Synthetic Division for g(7)

Next, perform synthetic division for \(g(7)\) : \[\begin{array}{c|ccccc} 7 & 1 & 0 & -4 & 0 & 3 & 2 \ & & 7 & 21 & 119 & 833 & 5843 \ \hline & 1 & 7 & 17 & 119 & 836 & 5845 \end{array}\] Thus, the value of \(g(7)\) is 5845.

Step 4: Synthetic Division for g(-1)

Finally, perform synthetic division for \(g(-1)\) : \[\begin{array}{c|ccccc} -1 & 1 & 0 & -4 & 0 & 3 & 2 \ & & -1 & 1 & 3 & -3 & 0 \ \hline & 1 & -1 & -3 & 3 & 0 & 2 \end{array}\] Thus, the value of \(g(-1)\) is 2. This completes all the synthetic divisions required.

Key Concepts

Polynomial Function Evaluation

Evaluating a polynomial function means finding the value of the polynomial for a specific value of its variable, usually denoted as x. In the context of college algebra, this concept is frequently practiced to understand the behavior of polynomial functions.

For example, given the function g(x) = x^6 - 4x^4 + 3x^2 + 2, we might be interested in finding the value of this function when x is 2, which can be denoted as g(2). A lengthy approach would be to plug 2 into the polynomial and calculate the powers and sums manually. However, synthetic division provides a shortcut, effectively reducing the time and effort needed to complete this calculation. When we perform synthetic division for g(2), the final number in the synthetic division tableau is 28, indicating that the value of the polynomial function g(x) when x is 2 is 28.

Understanding polynomial function evaluation is crucial as it helps in graphing functions, solving equations, and analyzing function behavior. By knowing how to evaluate polynomials efficiently, students can gain deeper insights into their properties and applications in various algebraic problems.

Polynomial Division

Polynomial division is a technique used to divide one polynomial by another, resulting in a quotient and sometimes a remainder. There are two main methods for dividing polynomials: long division and synthetic division. While long division provides a comprehensive view of the process, synthetic division is a simplified form employed when dividing by a linear factor of the form x - c.

Synthetic division is particularly useful when you have to evaluate a polynomial function at a specific value. For instance, when evaluating g(7) from the polynomial g(x) = x^6 - 4x^4 + 3x^2 + 2, synthetic division simplifies the process. We replace the x with 7 and rapidly compute the new coefficients by carrying out specialized subtraction and multiplication steps, which finally yield a value of 5845 for g(7). This method not only saves time but also minimizes potential calculation errors.

Understanding polynomial division is critical in algebra because it's used in solving higher-degree polynomial equations, simplifying rational expressions, and analyzing polynomial functions. Knowledge of this concept can significantly enhance a student's algebraic toolkit, leading to greater mathematical proficiency.

College Algebra

College algebra encompasses a wide range of topics that are foundational to mathematics and other sciences. It includes the study of functions, equations, inequalities, and various algebraic structures like polynomials and rational expressions.

Polynomial functions, which take the form an*x^n + ... + a2*x^2 + a1*x + a0 where n is a non-negative integer and a0, a1, ..., an are constants, are staple examples in college algebra courses. Synthetic division and function evaluation are techniques that help make working with these polynomials more manageable. These algebraic methods allow students to break down complex problems into simpler steps, thereby improving their analytical and problem-solving skills.

College algebra serves as the foundation for more advanced studies in mathematics and science. It equips students with the necessary tools for their academic and professional careers. From solving real-world problems to pursuing careers in the fields like engineering, economics, and computer science, the mastery of college algebra principles is indispensable. Therefore, developing a solid understanding of topics such as polynomial function evaluation and division through practice and application is essential for success in higher-level mathematics.