Problem 30 - Precalculus: Functions and Graphs

Question

Use synthetic division to find \(f(c)\). $$f(x)=-x^{3}+4 x^{2}+x ; \quad c=-2$$

Step-by-Step Solution

Answer

The value of \(f(-2)\) is 22.

Step 1: Set up the synthetic division

To use synthetic division, write down the coefficients of the polynomial in a row. For the polynomial \( f(x) = -x^3 + 4x^2 + x \), the coefficients are \([-1, 4, 1, 0]\). Also, write the value \(c = -2\) to the left of the coefficients.

Step 2: Bring down the leading coefficient

Bring down the leading coefficient, \(-1\), to the bottom row.

Step 3: Multiply and add

Multiply the number you just brought down (\(-1\)) by \(c = -2\), giving \(2\). Write \(2\) under the next coefficient (\(4\)). Add \(4 + 2 = 6\) and write \(6\) in the bottom row.

Step 4: Repeat the process

Multiply \(6\) by \(-2\), giving \(-12\). Write \(-12\) under the next coefficient (\(1\)). Add \(1 + (-12) = -11\) and write \(-11\) in the bottom row.

Step 5: Final multiplication and addition

Multiply \(-11\) by \(-2\), giving \(22\). Write \(22\) under the last coefficient (\(0\)). Add \(0 + 22 = 22\) and write \(22\) in the bottom row. This last number is the value of \(f(c)\).

Key Concepts

Polynomial Division

Polynomial division is a technique used to divide polynomials, similar to how we divide numbers. When we use polynomial division, we are dividing a polynomial, such as \(f(x) = -x^3 + 4x^2 + x\), by another binomial, usually in the form of \(x - c\). In this context, we're using a specific form of polynomial division called synthetic division.Using synthetic division helps simplify and solve polynomial equations quickly. It's a shortcut that's especially useful when you divide by a linear polynomial of the form \(x - c\). It's faster and more concise than long division of polynomials, and it's easier to manage without much paper work. To set up synthetic division, you line up the coefficients of the polynomial and perform a step-by-step process of multiplying and adding, as shown in the solved exercise above. This technique especially comes in handy when you want to evaluate the polynomial at a specific point, such as \(c = -2\) in the given example.

Function Evaluation

Function evaluation involves finding the output of a function for a specific input. In our example, we employ synthetic division to evaluate the function \(f(x) = -x^3 + 4x^2 + x\) at \(c = -2\). By using synthetic division, you not only determine the quotient but also immediately find the value of the polynomial at the given point, which is \(f(c)\).Function evaluation is essential because it lets us verify the output of a function without calculating the entire polynomial manually. This ability to check or find solutions quickly is a great asset when solving more complex mathematical problems.This process of evaluation is particularly effective when you suspect a specific value to potentially be a root or when you need to verify a result during problem-solving.

Roots of Polynomials

The roots of a polynomial are the values of \(x\) that make the polynomial equal to zero. Identifying these roots is often a primary goal when working with polynomials, as they provide critical points for graphing and analysis.Using synthetic division helps in finding these roots or confirming potential roots. In synthetic division, if the remainder (the last number you get from the process) is zero when evaluating at \(c\), that means \(c\) is indeed a root of the polynomial. In our case of evaluating \(f(x)\) at \(-2\), the final value was \(22\), suggesting that \(-2\) is not a root of the polynomial since it did not equal zero.Polynomial roots are important as they often represent significant moments in real-world scenarios, like points of intersection or change, making them essential for both algebraic calculations and applications.