Practice 1 — Divide \( x^3 - 8 \) by \( x - 2 \)
Answer
Coefficients of \( x^3 - 8 \) with placeholders: 1, 0, 0, -8; \( k = 2 \).
(This is the standard difference-of-cubes factorization \( x^3 - 8 = (x-2)(x^2 + 2x + 4) \).)
Overview
This interactive tool performs synthetic division of a polynomial \( P(x) \) by a linear factor \( (x - k) \). It computes the quotient \( Q(x) \) and remainder \( R \), and displays a full step-by-step synthetic division table.
The division identity is:
\[ \frac{P(x)}{x - k} = Q(x) + \frac{R}{x - k} \]where \( \deg(Q) = \deg(P) - 1 \) and \( R \) is a constant (the remainder).
Interactive Synthetic Division Tool
1. Enter the coefficients of the dividend in descending order, separated by commas.
Include zeros for missing powers.
2. Enter the value of \( k \).
3. Click Calculate.
Worked Example
Use synthetic division to divide \( P(x) = -x^5 + 6x^4 + 4x^2 + 4 \) by \( x - 3 \).
Step-by-step solution
Step 1 — Identify \( k \) and the coefficients.
The divisor is \( x - 3 \), so \( k = 3 \). Write the dividend with all descending powers, using zeros for missing terms:
\[ -1x^5 + 6x^4 + 0x^3 + 4x^2 + 0x + 4 \]Coefficients: -1, 6, 0, 4, 0, 4.
Step 2 — Set up and execute the synthetic division.
Drop the first coefficient, multiply by \( k \), add to the next column, repeat:
\[ \begin{array}{r|rrrrrr} 3 & -1 & 6 & 0 & 4 & 0 & 4 \\ & & -3 & 9 & 27 & 93 & 279 \\ \hline & -1 & 3 & 9 & 31 & 93 & 283 \end{array} \]Step 3 — Read off the quotient and remainder.
The bottom row gives the coefficients of \( Q(x) \), which has degree \( 5 - 1 = 4 \). The last entry is the remainder:
\[ Q(x) = -x^4 + 3x^3 + 9x^2 + 31x + 93 \] \[ R = 283 \]Check.
\[ (x - 3)(-x^4 + 3x^3 + 9x^2 + 31x + 93) + 283 = -x^5 + 6x^4 + 4x^2 + 4 \; ✓ \]Practice Exercises
Perform each division using the calculator, then check your work below.
Practice 2 — Divide \( 2x^4 + 3x^3 - 4x - 5 \) by \( x + 1 \)
Answer
Coefficients with placeholders: 2, 3, 0, -4, -5; \( k = -1 \).
Check: \( (x + 1)(2x^3 + x^2 - x - 3) + (-2) = 2x^4 + 3x^3 - 4x - 5 \) ✓
Practice 3 — Divide \( x^4 + 1 \) by \( x - 1 \)
(Demonstrates the need for placeholders.)
Answer
Coefficients with placeholders: 1, 0, 0, 0, 1; \( k = 1 \).
The remainder \( 2 \) also equals \( P(1) = 1^4 + 1 = 2 \) (Remainder Theorem).
Frequently Asked Questions
What is synthetic division?
Synthetic division is a shorthand method for dividing \(P(x)\) by a linear factor \(x - k\). It works only with coefficients. The result is a quotient \(Q(x)\) of degree \(\deg(P) - 1\) and a constant remainder \(R\), satisfying \(P(x)/(x - k) = Q(x) + R/(x - k)\).
When can I use synthetic division?
Only when the divisor is \(x - k\) (leading coefficient 1). For a divisor like \(2x + 3\), divide by \(x + 3/2\) first, then divide the quotient by 2. For higher-degree divisors, use polynomial long division.
What is the remainder in synthetic division?
The last number in the bottom row. By the Remainder Theorem, that value equals \(P(k)\), the polynomial evaluated at the divisor's root.
Why do I need to include zeros for missing powers?
Synthetic division assumes the coefficients are in descending order with no gaps. For \(P(x) = x^4 + 1\), enter 1, 0, 0, 0, 1. Skipping a power misaligns the columns and produces the wrong result.
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Last updated: October 17, 2026