Problem 45 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. If \(f(x)=2 x^{3}+3 x^{2}-19 x-4,\) and \(f(x)=(x+4) g(x)\) find \(g(x)\)

Step-by-Step Solution

Answer

The polynomial \(g(x)\) is \(2x^2 - 5x + 1\).

Step 1: Understand the Problem

We are given a polynomial function \( f(x) = 2x^3 + 3x^2 - 19x - 4 \) and told that it can be expressed as \( f(x) = (x+4)g(x) \). Our task is to find the polynomial \( g(x) \). This implies we need to perform polynomial division.

Step 2: Set Up Synthetic Division

Since \( x+4 \) is a factor of \( f(x) \), we will use synthetic division to divide \( f(x) \) by \( x+4 \). In synthetic division, \( x+4 \) corresponds to \( x = -4 \) as the root used in the setup.

Step 3: Perform Synthetic Division

We write the coefficients of \( f(x) \), which are 2, 3, -19, -4. Using the divisor \(-4\), perform synthetic division as follows:1. Bring down the leading coefficient 2.2. Multiply \(-4\) by 2, yielding -8, and add to the next coefficient 3, resulting in -5.3. Multiply \(-4\) by -5, yielding 20, and add to the next coefficient -19, resulting in 1.4. Multiply \(-4\) by 1, yielding -4, and add to the last coefficient -4, resulting in 0. The remainder is 0, confirming division.The quotient of the synthetic division is \(2x^2 - 5x + 1\).

Step 4: Write Down the Quotient as \(g(x)\)

The polynomial obtained from synthetic division, \(2x^2 - 5x + 1\), is \(g(x)\). Since the remainder is zero, \(f(x) = (x + 4)(2x^2 - 5x + 1)\), and thus \(g(x) = 2x^2 - 5x + 1\).

Key Concepts

Synthetic Division

Synthetic division is a simplified method of dividing polynomials, especially handy when dividing by a linear factor. It's faster and less cumbersome than long division. Unlike traditional division, synthetic division uses only the coefficients of the polynomial. If you have a polynomial like \(f(x) = 2x^3 + 3x^2 - 19x - 4\) and want to divide it by \(x + 4\), you substitute \(x = -4\) before you start.

Here's how it works in steps:

• First, write the coefficients of your polynomial that you want to divide: [2, 3, -19, -4].
• Place the root of the divisor \(x = -4\) to the left.
• Bring down the leading coefficient, which is 2 in this example.
• Multiply this leading coefficient by \(-4\) and add the result to the next coefficient. Continue this process for each coefficient.

The final sequence of numbers represents the coefficients of the quotient polynomial. If there's no remainder, this final number should be zero, confirming accurate division. Thus, synthetic division provides a fast and straightforward route to divide polynomials when you're dealing with linear divisors.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole number powers and coefficients. In general form, a polynomial in variable \(x\) looks like:
\[f(x) = a_n x^n + a_{n-1} x^{n-1} + \, ... \, + a_1 x + a_0\]
where \(a_n, a_{n-1}, \, ...\) are coefficients, \(n\) is a non-negative integer, and \(a_n eq 0\).

Understanding the main characteristics can help tackle various math problems:

• The degree of the polynomial is the highest power of \(x\).
• Leading coefficients affect the polynomial's behavior as \(x\) approaches infinity.
• Polynomial functions are continuous and smooth without any breaks or gaps.

Applications of polynomial functions are vast, ranging from modeling real-world phenomena to converting physical behaviors into mathematical equations. Whether handling revenue scenarios or processing signal data, understanding polynomial functions is invaluable.

Factoring Polynomials

Factoring polynomials is the process of expressing a polynomial as a product of its factors. It's a crucial technique that simplifies complex expressions, making them easier to solve or graph.

Consider the polynomial \(f(x) = 2x^3 + 3x^2 - 19x - 4\), which can be factored as \(f(x) = (x + 4)(2x^2 - 5x + 1)\).

Steps to factor a polynomial:

• Identify common factors in all terms. This simplifies the polynomial initially.
• Use techniques like synthetic division or the quadratic formula to further break down the polynomial.
• Check for zeros by substituting potential roots and verify with remainder zero from synthetic division.

Factoring polynomials is often the first step in solving polynomial equations, graphing them, or finding roots. Mastering techniques like synthetic division and recognizing common patterns aids in this essential mathematical skill.