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Divide by synthetic division.                         

The image you uploaded presents two division problems where you're required to use synthetic division. Let's break down the first problem.

### 1. Problem:
\[
\frac{6x^3 - 5x^2 + 4x - 1}{3x - 1}
\]

This is a division of a cubic polynomial by a linear binomial. To solve this using synthetic division, follow these steps:

#### Step 1: Prepare for synthetic division
- **Divisor:** First, rewrite the divisor \(3x - 1\) in the form \(x - c\). Here, divide the divisor by 3 to get \(x - \frac{1}{3}\).
- **Dividend coefficients:** The coefficients of the dividend polynomial \(6x^3 - 5x^2 + 4x - 1\) are \(6, -5, 4, -1\).

#### Step 2: Set up the synthetic division
We'll use \( \frac{1}{3} \) as the value of \(c\) and divide the coefficients step by step:

1. Bring down the first coefficient (6) unchanged.
2. Multiply 6 by \( \frac{1}{3} \) and add it to the second coefficient (-5).
3. Continue this process for all terms.

I'll solve this first problem and give a detailed explanation. Then I'll move on to the second one. Let's perform the calculation!The synthetic division of \( \frac{6x^3 - 5x^2 + 4x - 1}{3x - 1} \) gives the following results:

- **Quotient:** \( 6x^2 - 3x + 3 \)
- **Remainder:** 0

Thus, the exact division is:
\[
\frac{6x^3 - 5x^2 + 4x - 1}{3x - 1} = 6x^2 - 3x + 3
\]

Would you like me to proceed with solving the second problem, or provide more details on the first one?

Here are 5 questions related to synthetic division:
1. How do you handle division when the divisor has a leading coefficient greater than 1?
2. Can synthetic division be used when dividing by a quadratic binomial?
3. What happens if the remainder is not zero in synthetic division?
4. How do you set up synthetic division for a polynomial with missing terms?
5. How does synthetic division compare to long division?

**Tip:** Always double-check the setup of your synthetic division, especially the signs and divisor adjustment (like turning \(3x - 1\) into \(x - \frac{1}{3}\)).

Solve the polynomial division problem (6x^3 - 5x^2 + 4x - 1) ÷ (3x - 1) using synthetic division. This guide provides a step-by-step explanation, key formulas, and core concepts in polynomial division, suitable for students in Grades 10-12.

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