Complete The Synthetic Division Problem Below 2 8 6

Complete the Synthetic Division Problem Below: 2  8  6
A step‑by‑step guide to solving the division, understanding the method, and applying it to similar problems.

Introduction

Synthetic division is a streamlined version of polynomial long division that works specifically when dividing a polynomial by a linear factor of the form x − c. Because it eliminates the need to write variables and powers repeatedly, the process is faster, less error‑prone, and ideal for quickly finding quotients and remainders—or, equivalently, evaluating a polynomial at a given value (the Remainder Theorem).

In this article we will take the coefficient row 2  8  6, which corresponds to the quadratic polynomial

[ P(x)=2x^{2}+8x+6, ]

and complete a synthetic division problem. We will walk through two common scenarios—dividing by (x − 2) and by (x + 2)—so you can see how the same coefficient row yields different results depending on the chosen divisor. By the end, you’ll be able to:

• Set up a synthetic division table correctly.
• Perform the multiply‑add cycle with confidence.
• Interpret the quotient and remainder.
• Check your work using the Remainder Theorem or traditional long division.
• Avoid typical pitfalls that trip up beginners.

What Is Synthetic Division?

Synthetic division is a shortcut for dividing a polynomial P(x) by a linear binomial x − c. The algorithm uses only the coefficients of P(x) and the constant c (the zero of the divisor). The final row of numbers gives, from left to right, the coefficients of the quotient polynomial (one degree lower than the dividend) and the final value, which is the remainder R.

Mathematically, if

[\frac{P(x)}{x-c}=Q(x)+\frac{R}{x-c}, ]

then synthetic division produces Q(x) and R directly.

When to Use It

• The divisor is linear (degree 1) and of the form x − c.
• You need the quotient and/or remainder quickly (e.g., for factoring, solving equations, or evaluating P(c)).
• You prefer a compact, arithmetic‑only procedure over writing out variables and powers.

If the divisor is not linear (e.g., x² + 1) or is linear but not monic (e.g., 2x − 3), you must either factor out the leading coefficient or revert to long division.

Setting Up the Problem

Our dividend is 2x² + 8x + 6. The coefficient row, ordered from highest degree to constant, is:

2   8   6

We must decide on c. For illustration we will complete the synthetic division for two common choices:

1. Dividing by (x − 2) → c = 2
2. Dividing by (x + 2) → c = –2

(The sign change is crucial: x + 2 = x − (–2), so c = –2.)

Step‑by‑Step Synthetic Division

Below we detail the generic algorithm, then apply it to each case.

Generic Algorithm

1. Write the coefficients of the dividend in a row. 2. Place c (the zero of the divisor) to the left, often separated by a vertical bar.
2. Bring down the leading coefficient unchanged.
3. Multiply the value just written on the bottom row by c, write the product under the next coefficient.
4. Add the column, writing the sum in the bottom row.
5. Repeat steps 4‑5 until you have processed all coefficients.
6. The bottom row now contains:
   • All entries except the last → coefficients of the quotient (degree one less than the dividend).
   • The final entry → the remainder R.