Question

In Exercises 25–32, use synthetic division to evaluate the function for the indicated value of x. $$ f(x)=x^3-6 x+1 ; x=6 $$

Short answer

The value of the function \( f(x)=x^3-6x+1 \) for x = 6 is 1.

Step-by-step solution

Arrange in Standard Form

Before performing synthetic division, the polynomial \( f(x)=x^3-6x+1 \) should be arranged in standard form. In this case, it's already in standard form, that is, the terms are arranged in descending degree.

Identify Coefficients and Divider

Next, identify the coefficients of the polynomial and the value of x for which the function needs to be evaluated. The polynomial coefficients are 1 (for \(x^3\)), 0 (for \(x^2\)), -6 (for \(x\)) and 1 (for the constant); note the inclusion of a zero coefficient for \(x^2\) since it's not present in the polynomial. The value of x to use as the divider is 6.

Set Up Synthetic Division

Set up synthetic division by writing the coefficients and divider in appropriate places. Write the coefficients in a row and put the divider to the left, outside the vertical bar. It should look as follows: \n\n 6 | 1 0 -6 1 \n\n

Perform Synthetic Division

First, bring down the first coefficient and multiply it by the divider, placing the result under the second coefficient. Add these values and repeat the process with the new result. Continue until all coefficients have been processed. In this case:\n\n 6 | 1 0 -6 1 \n | 6 6 0\n --------------------\n 1 6 0 1 \n\n

Interpret the Result

The last number in the result row (the bottom row) is the function's value for the given x. In this case, the function \( f(x)=x^3-6x+1 \) evaluates to 1 when x = 6.

Key concepts

Polynomial Evaluation

Polynomial evaluation is a fundamental concept in algebra that involves finding the value of a polynomial function for a specific input. Given a polynomial, such as f(x) = x^3 - 6x + 1, evaluating it at x = 6 means calculating the value of the function when x is replaced with 6.

Understanding polynomial evaluation requires familiarity with the standard form of a polynomial and the ability to perform operations like addition, subtraction, and exponentiation. To evaluate f(6), replace every instance of x in the polynomial with 6 and simplify:
f(6) = (6)^3 - 6(6) + 1 = 216 - 36 + 1 = 181.

However, this direct calculation can be time-consuming, especially for higher degrees or more complex polynomials. This is where synthetic division can greatly simplify the process, particularly when you're only interested in the value of the polynomial for a given x, and not the entire function's behavior.

Standard Form of Polynomial

The standard form of a polynomial is an arrangement of terms in descending order by degree, where the degree of each term represents the exponent on the variable. Each term consists of a coefficient multiplied by the variable raised to an exponent.

For example, the polynomial f(x) = x^3 - 6x + 1 is written in standard form with its terms ordered from highest to lowest degree: the x^3 term first, followed by the implied 0x^2 term (which is not written out because its coefficient is zero), the -6x term, and finally the constant term +1.

It's important to recognize and align polynomials in standard form because it organizes the information in a way that makes it ready for synthetic division and other polynomial operations. For students grappling with these concepts, ensuring each term is in its proper place is the first essential step before any operation is performed on the polynomial.

Synthetic Division Steps

Synthetic division is a shorthand method of dividing a polynomial by a binomial of the form (x - c) and is particularly useful for evaluating polynomials at a specific point. The key advantage of synthetic division is that it is less labor-intensive than long division.

Here's an outline of the steps to perform synthetic division, which are reflected in the given example of f(x) = x^3 - 6x + 1 and x = 6:

• Write down the coefficients of the polynomial in descending order, including zero coefficients for any missing degrees.
• Write the number you're dividing by (in this case, 6) to the left of a vertical bar.
• Bring the first coefficient directly down below the horizontal line.
• Multiply this number by the divider and write the result under the next coefficient.
• Add the second coefficient to the result of that multiplication and write the sum below the line.
• Repeat the process of multiplying by the divider and adding vertically until all coefficients have been handled.
• The final number in the bottom row represents the value of the polynomial for the given x.

Following these steps for our example, the synthetic division table looks like this:
6 | 1 0 -6 1
| 6 6 0
----------------
1 6 0 1

As seen in the final row, the value on the far right is 1, which is the result of the evaluation f(6). Through practice, synthetic division can become a quick and efficient technique for polynomial evaluation.