Problem 39 - College Algebra

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. $$ x-2,3 x^{4}-6 x^{3}-5 x+10 $$

Step-by-Step Solution

Answer

Yes, \(x-2\) is a factor. Factorization: \((x-2)(3x^3-5)\).

Step 1: Set Up Synthetic Division

The divisor is of the form \(x - c\), so here \(c = 2\). Write down \(2\) and the coefficients of the polynomial \(3x^4-6x^3+0x^2-5x+10\) in order: \(3, -6, 0, -5, 10\).

Step 2: Bring Down the Leading Coefficient

Bring down the leading coefficient, which is \(3\), into the bottom row, as this will be the starting point for our synthetic division.

Step 3: Multiply and Add

Multiply the number obtained in the bottom row by \(2\) (the root) and write the result under the next coefficient.Then, add the column and write the result in the bottom row. Repeat this process across all columns: - Multiply \(3\) by \(2\): \(6\); add \(-6 + 6 = 0\). - Multiply \(0\) by \(2\): \(0\); add \(0 + 0 = 0\).- Multiply \(0\) by \(2\): \(0\); add \(-5 + 0 = -5\).- Multiply \(-5\) by \(2\): \(-10\); add \(10 + (-10) = 0\).

Step 4: Check the Remainder

The final number in the bottom row (from left to right) is the remainder of the division. Here, it is \(0\), indicating that \((x - 2)\) is a factor of \(3x^4 - 6x^3 - 5x + 10\).

Step 5: Write the Factorization

The numbers in the bottom row (except the last number) represent the coefficients of the resulting polynomial: \(3, 0, 0, -5\). This means:\[3x^3 + 0x^2 + 0x - 5 = 3x^3 - 5\]The factorization is therefore:\[(x - 2)(3x^3 - 5)\].

Key Concepts

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of simpler polynomials, which are called factors. In this exercise, we check if the expression \(x - 2\) is a factor of the polynomial \(3x^4 - 6x^3 - 5x + 10\). Factorization is useful in simplifying expressions and solving polynomial equations.

• The goal is to break down complex polynomials into simpler factors, making them easier to work with.
• Factors of a polynomial divide the polynomial exactly, meaning there's no remainder after division.

Applying synthetic division helps in identifying these factors quickly. It shows us whether the divisor is indeed a factor by resulting in a remainder of zero.

Divisor

In the context of synthetic division, a divisor is the expression or polynomial by which we divide the primary polynomial to determine if there's a factor.For our exercise, the divisor is \(x - 2\), a linear polynomial form. When using synthetic division, we focus on the constant part of the divisor, which is derived from \(x - c\).

• Here, \(c = 2\).
• This value is used throughout the synthetic division process.

Understanding the role of the divisor is crucial for carrying out synthetic division correctly. Using \(c\), we determine how the coefficients of the polynomial are manipulated during the process.

Remainder

The remainder in synthetic division is the final result when we divide a polynomial by a divisor. It tells us whether the divisor is a factor of the polynomial.

• If the remainder is zero, it confirms that the divisor is a factor.
• A non-zero remainder indicates that the divisor does not exactly divide the polynomial.

In our example, after completing synthetic division, the remainder was 0. This confirmed that \(x - 2\) is a factor of the polynomial, and we could thus write the factorization as \((x - 2)(3x^3 - 5)\). The remainder check is a crucial step in confirming factorization success.

Coefficients

Coefficients are the numerical factors in front of the variables in a polynomial. In synthetic division, these numerical values are essential for calculations.For the polynomial \(3x^4 - 6x^3 - 5x + 10\), the coefficients are listed in order:

• 3 (for \(x^4\))
• -6 (for \(x^3\))
• 0 (for \(x^2\), which is missing and hence represented by 0)
• -5 (for \(x\))
• 10 (constant term)

During synthetic division, we focus on these coefficients and apply the divisor's constant to compute the new set of coefficients for the reduced polynomial. In this problem, we find the new coefficients as \(3, 0, 0, -5\) after division, resulting in \(3x^3 - 5\). Understanding coefficients is vital to correctly handling polynomial manipulation and interpretation.