Problem 50 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. For what value of \(k\) is \(x+1\) a factor of \(f(x)=3 x^{4}+3 x^{3}+2 x^{2}+k x-4 ?\)

Step-by-Step Solution

Answer

The value of \(k\) is -2.

Step 1: Identify the Factor Theorem

According to the Factor Theorem, a polynomial function \(f(x)\) has \(x-a\) as a factor if and only if \(f(a) = 0\). In this problem, \(x+1\) is a factor, so we set \(x = -1\). This means \(f(-1) = 0\).

Step 2: Substitute and Simplify

Substitute \(x = -1\) into the function: \(f(-1) = 3(-1)^{4} + 3(-1)^{3} + 2(-1)^{2} + k(-1) - 4\). Simplify each term: \(3 \cdot 1 - 3 \cdot 1 + 2 \cdot 1 - k - 4\).

Step 3: Calculate the Expression

The expression from the previous step is simplified to: \(3 - 3 + 2 - k - 4\). Combine like terms: 1. \(3 - 3 = 0\)2. \(2 - 4 = -2\)Thus, the expression becomes \(-2 - k\).

Step 4: Solve for the Unknown Value

According to the Factor Theorem, \(-2 - k = 0\). Solve for \(k\) by adding 2 to both sides of the equation: \(-2 - k = 0\) gives \(-k = 2\). Multiply both sides by -1 to solve for \(k\): \(k = -2\).

Key Concepts

Understanding Polynomial Functions

Polynomial functions are mathematical expressions that consist of variables and coefficients, combined using the operations of addition, subtraction, multiplication, and non-negative integer exponents. A polynomial function of degree \(n\) can be written as:\[f(x) = a_n x^n + a_{n-1} x^{n-1} + \, \ldots \, + a_1 x + a_0\]Here, \(a_n, a_{n-1}, \ldots, a_0\) are constants, known as coefficients, and \(x\) is the variable. In our exercise, the polynomial function is \(f(x) = 3x^4 + 3x^3 + 2x^2 + kx - 4\).
Polynomials are used in various applications, from simple calculations to complex modeling in physics and engineering. Understanding polynomial functions is crucial because they form the backbone of many algebraic problems and solutions.

Exploring Factorization

Factorization involves breaking down numbers or expressions into their component parts, or 'factors,' which are multiplied together to obtain the original number or expression. In algebra, factorization helps simplify polynomial expressions or solve polynomial equations.
In our situation, we use the Factor Theorem, which states that a polynomial \(f(x)\) has \(x-a\) as a factor if and only if \(f(a) = 0\). This theorem is a powerful tool for polynomial factorization because:

• It allows you to test if a given expression is a factor.
• It provides a way to find missing coefficients in polynomials, as we did by finding \(k\).

Factorization reveals essential features of polynomials, such as roots and critical points, facilitating deeper understanding and problem-solving.

Working with Algebraic Expressions

Algebraic expressions are compositions of variables and numbers combined through mathematical operations like addition, subtraction, multiplication, and division. They serve as the building blocks for equations and inequalities in algebra.
When dealing with algebraic expressions, particularly in the context of polynomial functions, it's essential to understand the following:

• The concept of simplification, where expressions are reduced to their simplest form.
• How to substitute values into expressions, crucial for testing roots or solving equations, as seen in our exercise when substituting \(x = -1\) into \(f(x)\).
• Combining like terms effectively, which ensures clarity and simplicity, as was necessary to solve for the unknown \(k\) in the exercise.

Understanding and manipulating algebraic expressions augments our ability to tackle real-world problems efficiently. It forms a critical foundation for more advanced mathematical concepts and applications.