Problem 81 - Intermediate Algebra

Question

Use synthetic division to perform each division. $$ \frac{3 x^{3}-25 x^{2}+10 x-16}{x-8} $$

Step-by-Step Solution

Answer

The quotient is \(3x^2 - x + 2\).

Step 1: Identify the divisor and dividend

In synthetic division, we are dividing a polynomial by a linear divisor of the form \(x - c\). Here, the linear divisor is \(x - 8\) and the dividend (the polynomial) is \(3x^3 - 25x^2 + 10x - 16\).

Step 2: Set up the synthetic division

Write down the coefficients of the dividend: \(3, -25, 10, -16\). The constant \(c\) is \(8\) because our divisor is \(x - 8\).

Step 3: Begin synthetic division process

1. Bring down the first coefficient \(3\) to the bottom row.2. Multiply \(3\) by \(8\) (the constant), and write the result \(24\) under the next coefficient, \(-25\).3. Add \(-25 + 24\) to get \(-1\).

Step 4: Repeat the multiplication and addition steps

1. Multiply \(-1\) by \(8\), and write the result \(-8\) under the next coefficient, \(10\).2. Add \(10 + (-8)\) to get \(2\).3. Multiply \(2\) by \(8\), and write the result \(16\) under the last coefficient, \(-16\).4. Add \(-16 + 16\) to get \(0\), which confirms that there is no remainder.

Step 5: Write the quotient polynomial

Since we have a remainder of \(0\), the resulting coefficients of the polynomial (from left to right) are the coefficients of the resulting polynomial: \(3x^2 - x + 2\). This is the quotient of the division.

Key Concepts

Polynomial Division

Polynomial division is the process of dividing a polynomial by another polynomial, just like how you might divide numbers. In the division given here, we are using synthetic division to divide a cubic polynomial by a linear divisor. In polynomial division, our focus is on:

• Identifying the dividend, which is the polynomial being divided. In this exercise, it is the polynomial \(3x^3 - 25x^2 + 10x - 16\).
• Identifying the divisor, which specifies how we are dividing the polynomial. When using synthetic division, this divisor must be linear, meaning it is of the form \(x - c\).

To solve polynomial division problems using synthetic division, we work with the coefficients of the dividend; this simplifies the computation.

Linear Divisor

A linear divisor is a polynomial of the first degree. It is important in synthetic division due to its simplified form, \(x - c\), which makes calculation manageable. In our given problem, the linear divisor is \(x - 8\).Why use a linear divisor in synthetic division?

• The simplicity of a linear divisor, \(x - c\), allows us to use straightforward arithmetic instead of more complex methods that other polynomial degrees might require.
• It simplifies calculations by reducing the divisor to a single constant \(c\). This helps in focusing directly on manipulating coefficients.

A linear divisor is required for synthetic division because it allows the process to maintain an efficient structure, emphasizing its power in polynomial division.

Quotient Polynomial

The quotient polynomial is the result you get after dividing your dividend by your divisor. It represents the new polynomial formed, minus the remainder. Here,

• The process involves calculating each new coefficient by performing simple arithmetic operations: addition and multiplication.
• The coefficients from the synthetic division process lead to the formation of the final quotient polynomial.

In our example, after performing the synthetic division, the resulting quotient polynomial is \(3x^2 - x + 2\). Importantly, there was no remainder, indicating that the division was exact. Understanding the role of a quotient polynomial further aids in grasping polynomial division's applications in simplifying expressions.