Problem 339 - Precalculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Set up the Synthetic Division

To use synthetic division, we write the zero of the divisor on the left side. Since the expression is \(x - 2\), the zero is 2. Then, we write the coefficients of the dividend \(4x^3 - 3x^2 - 8x + 4\) in a row: 4, -3, -8, 4.

Step 2: Perform the Synthetic Division

Start by bringing down the leading coefficient (4). Multiply the zero (2) by this number and write the result under the next coefficient: \(2 \times 4 = 8\). Add this to the next coefficient (-3): \(-3 + 8 = 5\). Repeat this process: \(2 \times 5 = 10\), add to -8 to get 2; \(2 \times 2 = 4\), add to 4 to get 8.

Step 3: Determine the Remainder

The final number in the row is the remainder. In this case, it is 8. Since the remainder is not zero, \(x - 2\) is not a factor of \(4x^3 - 3x^2 - 8x + 4\).

Step 4: Conclusion on Factorization

Since the remainder is 8, \(x-2\) does not divide \(4x^3 - 3x^2 - 8x + 4\) evenly. Therefore, the first expression \(x-2\) is not a factor of the polynomial.

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another. It is similar to long division used in arithmetic. It's essential when simplifying expressions or solving polynomial equations.
There are several methods for polynomial division. Synthetic division is one such method, particularly useful when dividing a polynomial by a linear binomial, such as \(x - c\).

• It is a more straightforward and quicker method than long division for specific cases.
• This method reduces complexity by focusing on coefficients rather than full expressions.

The process requires knowledge of the divisor's zero, which is used to calculate results swiftly by iteratively carrying through coefficients. Understanding polynomial division is crucial for simplifying polynomials and solving equations.

Remainder Theorem

The Remainder Theorem plays a vital role in polynomial division. According to this theorem, when a polynomial \(f(x)\) is divided by \(x-c\), the remainder of this division is the same as \(f(c)\), the evaluation of the polynomial at \(x = c\).
This theorem helps determine whether a binomial \(x-c\) is a factor of the polynomial.

• If the remainder is zero, \(x-c\) is a factor.
• If the remainder is non-zero, \(x-c\) is not a factor.

In the synthetic division demonstration, the remainder was 8, indicating that \(x-2\) is not a factor of the polynomial \(4x^3 - 3x^2 - 8x + 4\). This knowledge helps in determining the roots and factorization of polynomials.

Factorization of Polynomials

Factorization of polynomials involves expressing the polynomial as a product of its factors. Each factor is simpler, often linear, helping in solving polynomial equations or simplifying expressions.
Finding these factors can be facilitated through techniques like polynomial and synthetic division, and applying the Remainder Theorem.

• Understanding factors helps in finding zeros of a polynomial.
• Simplified expressions arising from factorization can reveal more about the polynomial's behavior.

In our exercise, we found that \(x-2\) is not a factor of \(4x^3 - 3x^2 - 8x + 4\). This outcome implies that the polynomial cannot be broken down using \(x-2\) as one of its factors, highlighting the importance of factor calculations in polynomial analysis.