Problem 47 - College Algebra with Modeling and Visualization

Question

Use the remainder theorem to find the remainder when \(f(x)\) is divided by the given \(x-k\) $$f(x)=5 x^{2}-3 x+1 \quad\quad\quad x-1$$

Step-by-Step Solution

Answer

The remainder is 3.

Step 1: Identify the divisor

We are given the expression \(x-k\) where \(x-k = x-1\). This tells us that \(k = 1\).

Step 2: Apply the Remainder Theorem

According to the Remainder Theorem, the remainder when a polynomial \(f(x)\) is divided by \(x-k\) is \(f(k)\). Here, \(k = 1\). Thus, we need to calculate \(f(1)\).

Step 3: Evaluate the polynomial at \(k = 1\)

Substitute \(x = 1\) into the polynomial \(f(x) = 5x^2 - 3x + 1\).Calculate:\[f(1) = 5(1)^2 - 3(1) + 1\]\[= 5 - 3 + 1\]\[= 3\]

Step 4: State the remainder

Since \(f(1) = 3\), the remainder when \(f(x)\) is divided by \(x-1\) is 3.

Key Concepts

Polynomial Evaluation

Polynomial evaluation is a process where we determine the value of a polynomial expression at a certain point. In simple terms, it's about plugging a number into a polynomial and calculating the result. To evaluate a polynomial, follow these steps:

• Identify the specific value for which you need the evaluation (often designated as \(k\) in equations).
• Substitute this value into the polynomial for the variable \(x\).
• Perform the arithmetic operations to find the result.

For example, evaluating the polynomial \(f(x) = 5x^2 - 3x + 1\) at \(x = 1\) involves plugging \(1\) into the expression for \(x\), leading to \(f(1) = 5(1)^2 - 3(1) + 1 = 3\). This calculation gives us the output of the polynomial when \(x\) is \(1\), which is a crucial step in using the Remainder Theorem.

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a binomial of the form \(x - k\). It is faster than long division for polynomials and provides the quotient and remainder directly. The process can be outlined as follows:

• Write down the coefficients of the polynomial.
• Place the value of \(k\) outside this row of coefficients.
• Carry down the leading coefficient to a new row.
• Multiply \(k\) by this value and place the result under the next coefficient.
• Add this result to the coefficient directly above to fill the new row.
• Repeat this until the last coefficient is processed.

The last number in the final row represents the remainder. This method is particularly useful for quickly determining the remainder without fully dividing the polynomial. Though not applied in every Remainder Theorem problem, understanding synthetic division enriches comprehension of polynomial operations.

Polynomial Functions

Polynomial functions are a class of mathematical expressions that consist of variables raised to whole number powers and coefficients. The general form of a polynomial function is \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0\), where \(n\) is a non-negative integer, and each \(a_i\) is a coefficient.Key characteristics of polynomial functions include:

• The degree of a polynomial is the highest power of the variable \(x\). For instance, in \(5x^2 - 3x + 1\), the degree is 2.
• Polynomial functions are continuous and smooth graphs. They contain no breaks or sharp edges.
• These functions can model a variety of real-world phenomena because they can approximate more complex curves.

Understanding polynomial functions is essential for solving equations and analyzing mathematical relationships. They are foundational for algebra, calculus, and many applications in science and engineering.