Problem 42 - Algebra 2

Question

OPEN ENDED. Give an example of a polynomial function that has a remainder of 5 when divided by \(x-4 .\)

Step-by-Step Solution

Answer

The polynomial \( x^2 - 11 \) has a remainder of 5 when divided by \( x-4 \).

Step 1: Understand the Remainder Theorem

The Remainder Theorem states that the remainder of the division of a polynomial \( f(x) \) by a linear divisor \( x-a \) is equal to \( f(a) \). This means if \( f(x) \) is divided by \( x-4 \), then the remainder will be \( f(4) \).

Step 2: Set Up the Condition

From the exercise, we know that when a polynomial \( f(x) \) is divided by \( x-4 \), the remainder is 5. Hence, according to the Remainder Theorem, \( f(4) = 5 \).

Step 3: Construct the Polynomial Function

To construct a polynomial satisfying this condition, we choose a simple polynomial like \( f(x) = x^2 \). Calculate \( f(4) = 4^2 = 16 \). Since \( 16 eq 5 \), we adjust the polynomial. Let's try \( f(x) = x^2 - 11 \) as a second example:Calculate \( f(4) = 4^2 - 11 = 16 - 11 = 5 \). This satisfies the condition.

Step 4: Verify the Solution

Substitute \( x=4 \) back into the function \( f(x) = x^2 - 11 \):\( f(4) = 16 - 11 = 5 \), which matches the remainder given in the problem statement. Therefore, the polynomial \( f(x) = x^2 - 11 \) is a valid solution.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions made up of variables raised to whole number powers, multiplied by coefficients and summed together. They're versatile, used to model various real-world phenomena.

Key characteristics of polynomial functions include:

• They are continuous and smooth, with no sharp corners or gaps.
• The degree of the polynomial, which is the highest power of the variable, dictates the 'shape' or 'turning points' of the graph.
• Coefficients can be any real number, making an infinite variety of polynomial forms.

For example, the polynomial function used in our problem, such as
\(f(x) = x^2 - 11\), is a quadratic polynomial because it has a degree of 2. This kind of polynomial forms a parabola when graphed. Understanding polynomials forms the basis for exploring various algebraic concepts like the ones involved in our exercise.

Linear Divisor

A linear divisor is a simple type of divisor in polynomial division. It takes the form of \(x-a\), where \(a\) is a constant. This linear form means the divisor graphically represents a straight line intersecting the x-axis at \(x=a\).

The role of a linear divisor in polynomial division is key:

• It simplifies the division process due to its straightforward form.
• The x-intercept of this linear function helps locate roots of the polynomial when the complete division results in no remainder.

In our exercise, the linear divisor is \(x-4\). Using this in our division allows us to focus on the value of the polynomial at \(x=4\), aiding in the application of the Remainder Theorem. Recognizing and working with linear divisors is foundational to higher mathematics, offering a stepping stone to more complex algebraic operations.

Remainder Theorem Application

The Remainder Theorem is a fundamental tool in algebra for simplifying the division of polynomials. It directly relates the remainder of polynomial division to evaluating the polynomial function at a point.

Here's a breakdown of how the theorem works:

• If a polynomial \(f(x)\) is divided by \(x-a\), the remainder of this division is \(f(a)\).
• This simplifies finding the remainder without performing full polynomial division, which can be time-consuming.

In our example, we used the Remainder Theorem to determine that when the polynomial \(f(x)\) is divided by the linear divisor \(x-4\), the remainder is 5. By evaluating \(f(4)\) and setting it equal to 5, we were able to adjust a simple polynomial to meet this condition. This concept is not only helpful in homework but also provides a practical method widely applicable in solving real-world problems involving polynomial functions.