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Let's dive into the concept of algebra, which is the foundation of synthetic division. Algebra involves using symbols and letters to represent numbers and quantities in formulas and equations. This branch of mathematics is crucial for solving various types of mathematical problems, including polynomial division. Polynomial division is a type of algebraic operation where we divide a polynomial by another polynomial. In our exercise, we need to divide the polynomial \[5x^3 - 6x^2 + 3x + 14\] by \[x + 1\] using synthetic division.

So, why synthetic division? It's a shortcut method that makes the division process simpler and quicker compared to long polynomial division.

To start, we need to understand that synthetic division can only be used when dividing by a linear divisor. A linear divisor has the form \[x - c\]. Here, our divisor is \[x + 1\], which can be rewritten as \[x - (-1)\]. Notice how the root of the divisor changes the sign from positive to negative.

We then write down the coefficients of the polynomial, which are 5, -6, 3, and 14. These values represent the terms \(5x^3\), \(-6x^2\), \(3x\), and the constant 14, respectively. In synthetic division, we also need the root of the divisor, which in this case is -1. From there, it's a matter of following systematic steps to achieve the result.

Polynomial division, much like regular division, involves distributing terms in the numerator (dividend) by the terms in the denominator (divisor). Let's use synthetic division to handle our polynomial \[5x^3 - 6x^2 + 3x + 14\].

Here are the steps for synthetic division:

• Write down the coefficients of the polynomial: 5, -6, 3, 14.
• Write the root of the divisor on the left. Since we are dividing by \(x + 1\), the root is -1.
• Bring down the leading coefficient (5) directly below the line.
• Multiply the brought-down number (5) by the root (-1) and place the result below the next coefficient (-6).
• Add the coefficients together: -6 + (-5) = -11.
• Repeat the multiplication and addition steps for the rest of the coefficients.

Through these steps, we turn polynomial division into a simple process of multiplication and addition. This makes it more efficient and less prone to errors than traditional long division.

In the context of polynomial division, the remainder plays a crucial role in determining the final result of the division. In our example, after performing synthetic division on \[5x^3 - 6x^2 + 3x + 14\] by \[x + 1\], the remainder is zero.

To find the remainder using synthetic division:

• Perform the multiplication and addition steps for each coefficient separately.
• The last value obtained after all calculations represent the remainder.
• If the remainder is zero, it means the polynomial divides evenly by the divisor.

In this case, the last addition resulted in zero, indicating that \[x + 1\] is a factor of the polynomial \[5x^3 - 6x^2 + 3x + 14\]. Therefore, the quotient of the division is \[5x^2 - 11x + 14\], and there’s no leftover value or remainder.