Problem 36 - College Algebra

Question

Given \(f(x)=3 x^{4}+6 x^{3}-2 x+4,\) use the Remainder Theorem to find \(f(-4)\).

Step-by-Step Solution

Answer

The value of \(f(-4)\) is 396.

Step 1: Substitute \(x\) with \(-4\)

Replace every instance of \(x\) in the polynomial function \(f(x)=3 x^{4}+6 x^{3}-2 x+4\) with \(-4\) to get: \(f(-4)\) = \(3*(-4)^4 + 6*(-4)^3 - 2*(-4) + 4\).

Step 2: Simplify the equation

Simplify the equation to get the final value. This involves performing the exponentiation first (according to the order of operations, or BIDMAS/BODMAS), followed by multiplication or division, and finally, addition or subtraction. So, \(f(-4) = 3*256 + 6*(-64) - 2*(-4) + 4 = 768 - 384 + 8 + 4\).

Step 3: Compute the final Result

Add and subtract the values to find \(f(-4)\): \(f(-4) = 768 + 8 + 4 - 384 = 396\).

Key Concepts

Polynomial Evaluation

Evaluating polynomials is a straightforward process where you substitute a specific value for the variable and calculate the result. In the given problem, we have the polynomial function \(f(x) = 3x^4 + 6x^3 - 2x + 4\). To evaluate \(f(-4)\), replace every \(x\) with \(-4\). You will perform this operation with care:

• Start by substituting every instance of \(x\) with \(-4\).
• This transforms the function into a numerical expression \(3(-4)^4 + 6(-4)^3 - 2(-4) + 4\).
• Make sure to follow the correct order of operations to simplify it effectively.

Each step in the evaluation must be carried out carefully to ensure accuracy in calculation.

Order of Operations

The order of operations is crucial to solving algebraic expressions correctly. The acronym BIDMAS/BODMAS can help you remember this sequence: Brackets, Indices (Exponents), Division/Multiplication, Addition/Subtraction. When evaluating \(f(-4) = 3(-4)^4 + 6(-4)^3 - 2(-4) + 4\), follow these rules:

• First, calculate the powers: Compute \((-4)^4=256\) and \((-4)^3=-64\).
• Next, multiplication: Perform multiplications: \(3 imes 256\), \(6 imes -64\), and \(-2 imes -4\).
• Then, addition and subtraction: Combine all these values by adding and subtracting to get the final result.

The priority of operations is fundamental to obtaining the correct outcome for any polynomial evaluation.

Algebraic Substitution

Algebraic substitution allows us to explore the behavior of polynomial expressions for different inputs. In this context, it involves replacing variable \(x\) with specific numbers, like \(-4\) in \(f(x) = 3x^4 + 6x^3 - 2x + 4\). This requires careful attention since:

• Every occurrence of \(x\) must be replaced: Ensure no instance is overlooked.
• Produce a new expression: This gives a number-only equation based on the original polynomial.
• Helps in finding specific values: Such substitutions determine function values at specific points, like finding \(f(-4)\).

Algebraic substitution is a powerful tool, especially when applying concepts like the Remainder Theorem, to simplify complex algebraic functions.