Problem 34 - College Algebra

Question

Given \(f(x)=x^{3}-7 x^{2}+5 x-6,\) use the Remainder Theorem to find \(f(3)\).

Step-by-Step Solution

Answer

The value of the function \(f(x)\) when \(x=3\) is -27.

Step 1: Understand the Remainder Theorem

The Remainder Theorem states that the remainder of the division of a polynomial \(P(x)\) by the binomial \(x-a\) is \(P(a)\). Therefore, to find \(f(3)\), it is necessary to substitute \(x\) with 3 into the equation.

Step 2: Substitute x with 3

Substitute \(x\) with 3 into the function \(f(x) = x^{3} - 7x^{2} + 5x - 6\) to get \(f(3) = 3^{3} - 7*(3)^{2} + 5*3 - 6\).

Step 3: Perform the mathematical operations

Perform the mathematical operations to determine the value of \(f(3)\). Thus, \(f(3) = 27 - 63 + 15 - 6 = -27\).

Key Concepts

Polynomial Division

Polynomial division is similar to the long division that many are familiar with from elementary school. However, instead of dividing numbers, we divide polynomial expressions. The core idea is to determine how many times a polynomial, called the divisor, fits into another one, the dividend, producing a quotient and sometimes a remainder.
The Remainder Theorem is a helpful tool in this process, as it simplifies the task of finding remainders when dividing by binomials of the form \(x-a\). You only need to substitute \(a\) into the polynomial to get the remainder. This eliminates the need for lengthy division processes, speeding up your calculations.

Here's a quick summary of the division steps:

• Identify the divisor and dividend.
• Use multiplication and subtraction to simplify the dividend, gradually breaking it down step by step.
• Repeat until the degree of the remainder is less than the degree of the divisor.

Using the Remainder Theorem streamlines this process, allowing for quicker solutions when dealing with binomials.

Evaluate Polynomial

Evaluating a polynomial involves determining its value for a specific variable. In this context, we want to find the value of \(f(x)\) when \(x\) is set at a certain point, like \(x = 3\) in the given exercise.
This process requires replacing the variable in the polynomial with the specified number and then calculating the value. When you evaluate polynomials, each term's power of \(x\) matters significantly, as it affects the term's contribution to the polynomial's overall value.

To be more precise when evaluating the polynomial \(f(x) = x^3 -7x^2 + 5x - 6\) for \(x=3\):

• Calculate each term independently: \(3^3 = 27\), \(-7 \times 3^2 = -63\), \(5 \times 3 = 15\), and \(-6\).
• Add these values together: \(27 - 63 + 15 - 6 = -27\).

This results in \(f(3) = -27\). Evaluating such polynomials repeatedly benefits from understanding how to handle powers and multiplication accurately.

Substitute Value in Polynomial

Substitution is a simple yet powerful mathematical technique. It involves replacing a variable in an equation with a specified value. In the given exercise, this means substituting \(x\) with \(3\) in the polynomial \(f(x) = x^3 - 7x^2 + 5x - 6\).
The substitution process transforms the polynomial from an expression involving \(x\) into an arithmetic problem with specific numbers, making it easier to evaluate. By assigning a specific number to \(x\), you simplify the expression down to basic computation.

When substituting:

• Carefully replace every instance of \(x\) in the polynomial with the number chosen (\(3\) in this case).
• Recalculate each term based on this new number.

As shown, substituting helps convert abstract polynomial expressions into tangible numbers that can be managed easily, paving the way for straightforward solutions using basic arithmetic steps.