Problem 38 - College Algebra with Corequisite Support

Question

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. $$ x-2, \quad 4 x^{3}-3 x^{2}-8 x+ 4 $$

Step-by-Step Solution

Answer

\(x - 2\) is not a factor since the remainder is 8.

Step 1: Identify the divisor and dividend

We need to check if the polynomial \(x - 2\) is a factor of the polynomial \(4x^3 - 3x^2 - 8x + 4\). Here, the divisor is \(x-2\) and the dividend is \(4x^3 - 3x^2 - 8x + 4\).

Step 2: Set up synthetic division

To use synthetic division with \(x - 2\), we need the root of the divisor, which is \(x = 2\). We write this root at the side and list the coefficients of the dividend polynomial: 4, -3, -8, and 4.

Step 3: Perform synthetic division - First step

Bring the first coefficient (4) down unchanged to the bottom row. It becomes the first term of the quotient.

Step 4: Perform synthetic division - Iterate through coefficients

Multiply the bottom number by the root (2), then add the result to the next coefficient: 4 * 2 = 8, -3 + 8 = 5, so write 5 in the bottom row.

Step 5: Continue synthetic division

Repeat the process: 5 * 2 = 10, next coefficient -8 + 10 = 2, write 2 below.

Step 6: Final synthetic division step

Continue: 2 * 2 = 4, final coefficient 4 + 4 = 8, write 8 below.

Step 7: Analyze the remainder

The final number in the bottom row is the remainder. Here, the remainder is 8, which means \(x - 2\) is not a factor of \(4x^3 - 3x^2 - 8x + 4\).

Key Concepts

Polynomial Factorization

Polynomial factorization is a crucial concept in algebra that involves expressing a polynomial as a product of its factors. These factors can be simpler polynomials or constants. The fundamental goal here is to simplify polynomial expressions to their most basic components. Factorization makes solving polynomial equations easier because it breaks down complex expressions into manageable parts.

In this exercise, you were asked to determine if the polynomial \(x - 2\) is a factor of \(4x^3 - 3x^2 - 8x + 4\). A factor of a polynomial is another polynomial that divides the original polynomial without leaving a remainder.

• If a polynomial has a factor, it means you can express the polynomial as the product of that factor and another polynomial.
• In simpler terms, factoring helps in breaking down a polynomial into smaller 'pieces' which, when multiplied, result in the original polynomial.

Understanding factorization lays the groundwork for more complex algebraic operations and aids in solving polynomial equations.

Remainder Theorem

The Remainder Theorem is a powerful tool in algebra that connects division and factorization. According to this theorem, when a polynomial \(f(x)\) is divided by a linear divisor \(x-a\), the remainder of this division is \(f(a)\).

Let’s see how this relates to the exercise performed. By using synthetic division, we tested if \(x - 2\) was a factor of \(4x^3 - 3x^2 - 8x + 4\). After performing synthetic division, the remainder was found to be 8. According to the Remainder Theorem, this tells us that when the polynomial is evaluated at \(x = 2\), it gives us a remainder of 8, meaning \(x - 2\) is not a factor of the polynomial.

• If the remainder turns out to be zero, then \(x - a\) is a factor of the polynomial.
• The theorem helps quickly determine factor relationships without fully dividing polynomials, saving time and effort.

Divisors and Dividends

In terms of polynomials, the divisor and dividend are analogous to these terms in numerical division. The **divisor** is the expression you divide by, while the **dividend** is the expression being divided.

In our exercise, you checked if \(x - 2\) (the divisor) divided perfectly into \(4x^3 - 3x^2 - 8x + 4\) (the dividend). This resulted in a non-zero remainder, demonstrating that the divisor does not evenly divide the dividend.

• The result of the division, if the remainder is zero, reveals polynomial factorization.
• Understanding the roles of divisor and dividend is crucial and simplifies polynomial division methods like synthetic division.

Gain familiarity with these terms as they frequently appear in more complex algebraic manipulations.

Polynomial Division Steps

Performing polynomial division, especially synthetic division, involves systematic steps to avoid confusion and errors. Synthetic division is often preferred for its efficiency, particularly with linear divisors.

Here's a simplified sequence of synthetic division, as demonstrated:

• Identify the divisor and write out its root. For \(x - 2\), the root is 2.
• Record coefficients of the dividend. List them in sequence: 4, -3, -8, and 4.
• Begin division. Bring down the leading coefficient.
• Iterate the multiplication and addition process. Multiply the sum so far by the divisor’s root and add it to the next coefficient.
• Continue until all coefficients are processed. The final numbers give the quotient and the remainder.

In our case, the process yielded a remainder of 8, indicating that \(x - 2\) does not divide \(4x^3 - 3x^2 - 8x + 4\) evenly. This highlights the necessity of following each procedural step carefully to arrive at correct conclusions.