Problem 23 - Precalculus : Building Concepts and Connections

Question

Use synthetic division to find the function values. \(f(x)=-2 x^{4}-10 x^{3}-3 x+10 ;\) find \(f(-1)\) and \(f(2)\)

Step-by-Step Solution

Answer

The function values are \(f(-1) = 21\) and \(f(2) = -108\).

Step 1 : Set Up Synthetic Division to find f(-1)

Set up the Synthetic Division by placing the coefficients of the polynomial at the top of your synthetic division structure, -2, -10, 0 (since there is no \(x^2\) term, we add 0), -3, and 10. Then you place the value of x you are trying to evaluate, which is -1, on the outside of your synthetic division structure.

Step 2 : Carry Out Synthetic Division for f(-1)

Drop down the first coefficient, -2, untouched. Multiply -1 by -2 to get 2 and add this to -10 to get -8. Continue the process: multiply -1 by -8 to get 8 and add this to 0 to get 8. Multiply -1 by 8 to get -8 and add this to -3 to get -11. Multiply -1 by -11 to get 11 and add this to 10 to get 21.

Step 3 : Interpret the Result for f(-1)

The last number you get, 21, is the value of the function at x=-1. So, \(f(-1) = 21\).

Step 4 : Set Up Synthetic Division to find f(2)

Repeat the process for \(x = 2\). Set up the Synthetic Division by placing the coefficients of the polynomial at the top of your synthetic division structure, -2, -10, 0 (since there is no \(x^2\) term, we add 0), -3, and 10. Then you place the value of x you are trying to evaluate, which is 2, on the outside of your synthetic division structure.

Step 5 : Carry Out Synthetic Division for f(2)

Drop down the first coefficient, -2, untouched. Multiply 2 by -2 to get -4 and add this to -10 to get -14. Continue the process: multiply 2 by -14 to get -28 and add this to 0 to get -28. Multiply 2 by -28 to get -56 and add this to -3 to get -59. Multiply 2 by -59 to get -118 and add this to 10 to get -108.

Step 6 : Interpret the Result for f(2)

The last number you get, -108, is the value of the function at x=2. So, \(f(2) = -108\).

Key Concepts

Polynomial Evaluation

Evaluating polynomials is an important skill, especially when using synthetic division. It involves finding the value of a polynomial at a specific point. This can be done using synthetic division, as it provides a quick and efficient method.

• Start by writing down the coefficients of the polynomial.
• If there are missing terms, fill them with zero.
• Use synthetic division, which simplifies calculations compared to traditional methods such as direct substitution.

For example, with the polynomial \(-2x^4 - 10x^3 - 3x + 10\), to evaluate \(f(-1)\), use the coefficients: \(-2, -10, 0, -3,\) and \(10\). Then apply synthetic division with \(x = -1\) to find the function value.

Function Value

The function value, \(f(x)\), represents the output of a function given an input \(x\). To determine \(f(x)\) using synthetic division:

• Identify the input value you want to test (e.g., \(-1\) or \(2\)).
• Perform synthetic division using this input.
• The last number you obtain from synthetic division gives \(f(x)\).

For instance, when evaluating a polynomial \(-2x^4 - 10x^3 - 3x + 10\) at \(x = -1\), the function value is \(21\). Similarly, at \(x = 2\), the function value is \(-108\). This provides a clear understanding of how the function behaves at these points.

Roots of Polynomials

Understanding the roots of polynomials is crucial, as they are the values of \(x\) that make the polynomial equal to zero. Synthetic division helps in not only evaluating polynomials but also in finding their roots.

• If the remainder of the synthetic division is zero, then the test value is a root.
• No remainder indicates that the polynomial can be factored by \((x - \text{test value})\).
• If you get a non-zero remainder, the test value is not a root.

In the examples \(f(-1) = 21\) and \(f(2) = -108\), neither \(-1\) nor \(2\) is a root of the polynomial \(-2x^4 - 10x^3 - 3x + 10\), since the remainders were not zero. Thus, neither is a solution to \(f(x) = 0\).