Problem 33 - College Algebra

Question

Given \(f(x)=2 x^{3}-11 x^{2}+7 x-5,\) use the Remainder Theorem to find \(f(4)\)

Step-by-Step Solution

Answer

The value of function \(f(x)=2 x^{3}-11 x^{2}+7 x-5\) for \(x=4\) is 147.

Step 1: Substitution

Substitute \(x=4\) into the function \(f(x)=2 x^{3}-11 x^{2}+7 x-5\). Thus the function becomes, \(f(4)=2(4)^{3}-11(4)^{2}+7(4)-5\).

Step 2: Simplification

Simplify the expression to get the value of \(f(4)\). In this step, follow the order of operations-BODMAS (Brackets, Orders, Division and Multiplication, and Addition and Subtraction). Thus, the calculation of the expression will be as follows:\n \(f(4)=2(64)-11(16)+7(4)-5\),\n \(f(4)=128-176+28-5\),\n \(f(4)=152-5\),\n \(f(4)=147\)

Key Concepts

Polynomial Functions

Polynomial functions are an essential type of mathematical function involving variables raised to various powers and multiplied by coefficients. Specifically, a polynomial function is expressed as a sum of terms, each of which is a constant multiplied by a variable raised to a non-negative integer power. This type of function is often written in the form:

• The equation: \( a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \)
• Where \( a_n, a_{n-1}, ..., a_0 \) are constants, and \( n \) is a non-negative integer.

In our exercise, we have the polynomial function \( f(x) = 2 x^{3} - 11 x^{2} + 7 x - 5 \), where each term has a coefficient and a power of \( x \). The Remainder Theorem allows us to evaluate this function at a specific point easily, and we'll explore that through substitution.

Substitution Method

The substitution method is a straightforward approach used to find the value of polynomial functions at specific points. By substituting the given value of the variable into the function, we can directly evaluate the polynomial. To apply the substitution method in the context of our exercise:

• Identify the polynomial function: \( f(x) = 2 x^{3} - 11 x^{2} + 7 x - 5 \).
• Substitute \( x = 4 \) into the polynomial: \( f(4) = 2(4)^{3} - 11(4)^{2} + 7(4) - 5 \).

Performing substitution involves replacing each \( x \) in the polynomial with 4, transforming the expression into one that can be readily simplified.

Order of Operations

The order of operations is a guideline used in mathematics to determine the sequence in which operations are performed, ensuring consistency in calculations. This rule is often remembered by the acronym BODMAS, which stands for:

• Brackets
• Orders (i.e., powers and roots)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

In the context of evaluating the polynomial at \( x = 4 \):

• First, calculate the powers: \( 2(4)^{3} = 2(64) \) and \( -11(4)^{2} = -11(16) \).
• Multiply the powers by their coefficients.
• Finally, perform addition and subtraction in the order they appear to simplify \( f(4) = 128 - 176 + 28 - 5 \).

Function Evaluation

Function evaluation is the process of finding the output of a function for a specific input. It involves substituting the input value into the formula of the function and calculating the result. In using the Remainder Theorem, function evaluation helps determine whether a certain value is a zero of the polynomial or it calculates the remainder when divided by \( x-c \).In our specific example:

• The function \( f(x) \) is evaluated at \( x = 4 \).
• We substitute and follow the order of operations to simplify: \( f(4) = 147 \).
• The result 147 is the function's output, providing insight into the behavior of the polynomial at \( x = 4 \).

This evaluation process is a vital skill in analyzing and understanding polynomial functions.