Question

In Exercises 25–32, use synthetic division to evaluate the function for the indicated value of x. $$ f(x)=x^3-2 x^2+4 x+3 ; x=2 $$

Short answer

The function \(f(x) = x^3-2 x^2+4 x+3\) evaluated at \(x = 2\) is \(11\).

Step-by-step solution

Set up Synthetic Division

Set up the synthetic division tableau. Write down the coefficients of \(f(x)\) in descending order of power of x on the top row. Place \(x=2\) outside the tableau.

Perform Synthetic Division

Carry the leading coefficient \(1\) straight down, then multiply it by \(2\) (the chosen value for \(x\)) and write the product under the next coefficient \(-2\). Add \(-2\) and \(2*1=-2\) to get \(0\). Multiply again by \(2\) to get \(0\), write it under the third coefficient \(4\), add them to get \(4\). Perform one more time to get \(2*4=8\) under \(3\). Sum to get \(11\). The last row represents the coefficients of the division result.

Evaluate the Function

The last number in the bottom row of the tableau is the result of the function evaluated at \(x=2\). Therefore, \(f(2) = 11\).

Key concepts

Evaluating a Function

When you evaluate a function, you’re essentially substituting a specific value into the function in place of the variable. Imagine a function as a mathematical vending machine; when you input a number (like pressing a button for a snack), the machine processes it according to its internal settings (the algebraic formula), and out comes your result (the value of the function).

For instance, with the function in the exercise, you want to find out what the machine gives you when you input the number 2. Symbolically, you're looking for the output of the function when it sees an 'x' value of 2, which is notated as \(f(2)\). By plugging in 2 wherever there's an 'x' in the equation, you determine the corresponding output of the function.

Polynomial Division

The method of synthetic division is a shorthand, or simplified, procedure for dividing a polynomial by a binomial of the form \(x - c\). It’s especially handy when you want to evaluate a polynomial function for a specific value of 'x', much like in our given problem.

Polynomial division might sound daunting, but it’s just breaking down a larger expression into a more manageable one. It’s akin to sharing a pizza evenly among friends. If you cut a pizza into smaller slices, each person gets a piece; in polynomial division, you break down the terms of a polynomial until what remains is a collection of simpler expressions (the slices), and possibly a remainder.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operation symbols that represents a mathematical relationship. Variables in these expressions are like wildcards in a game—they can stand in for unknown values. When you perform operations on algebraic expressions such as addition, subtraction, multiplication, and division, you’re playing with these wildcards to figure out what they represent.

In the context of the exercise, the algebraic expression is \(f(x)=x^3-2 x^2+4 x+3\), and our focus is on evaluating this expression when x equals 2. Think of it as giving a specific role to your wildcard in the game—once you’ve set it, you can follow through the operations to find out how all the other pieces fit together.