Math & Algebra

Free Polynomial Solver & Synthetic Division Calculator

Solving higher-degree polynomial equations and evaluating algebraic functions at specific values is a foundational task in algebra, pre-calculus, and calculus. Synthetic division simplifies polynomial long division when dividing by linear binomials (x - k). Our free polynomial calculator lets you perform synthetic division calculator steps, solve polynomial roots solver questions, and compute cubic equation solver step by step evaluations.

Polynomial Solver & Synthetic Division Calculator

Evaluate cubic polynomials $f(x) = ax^3 + bx^2 + cx + d$, calculate derivatives, and perform synthetic division.

Coeff a (x³)

Coeff b (x²)

Coeff c (x¹)

Coeff d (Constant)

Evaluation & Division Point (x = k):

Polynomial Value f(2)

x = 2 is a ROOT of f(x) (Remainder is 0).

First Derivative f'(x):3x² - 12x + 11

Quotient Degree:Degree 2 (Quadratic)

Synthetic Division Table: f(x) ÷ (x - 2)

2 \|	1	-6	11	-6
	↓ 0	+2	+-8	+6
Result	1	-4	3	0

Quotient Polynomial Q(x):

Q(x) = 1x² - 4x + 3 with Remainder = 0

Mathematical Formulas & Calculation Methods

Cubic Polynomial & Synthetic Division Formula

Polynomial f(x) = ax³ + bx² + cx + d evaluated at x = k yields quotient Q(x) and remainder R.

f(x) = (x - k) × Q(x) + R | Synthetic Row: [a, b + k*a, c + k*(b+k*a), d + k*c_new]

Worked Step-by-Step Examples

Example 1: Divide f(x) = x³ - 6x² + 11x - 6 by (x - 2)

"Perform synthetic division with x = 2."

Steps:

Row 1 Coeffs: [1, -6, 11, -6].
Step 1: Bring down 1.
Step 2: 1 × 2 = 2 → -6 + 2 = -4.
Step 3: -4 × 2 = -8 → 11 + (-8) = 3.
Step 4: 3 × 2 = 6 → -6 + 6 = 0 (Remainder = 0).
Quotient Q(x) = x² - 4x + 3.

Result: Q(x) = x² - 4x + 3, Remainder = 0 (x = 2 is a root!).

What is Polynomial Solver & Synthetic Division Calculator?

A synthetic division calculator is a streamlined algebraic tool for dividing polynomials by linear binomials without full long division.

The Remainder Theorem states that evaluating f(k) gives the exact remainder when f(x) is divided by (x - k).

How to Use Polynomial Solver & Synthetic Division Calculator

1Enter polynomial coefficients a (x³), b (x²), c (x¹), and d (constant).
2Enter evaluation/division point x = k.
3Review the full synthetic division step table, quotient equation, derivative, and remainder.

Key Features

Generates clear synthetic division table matrix.
Displays quotient polynomial Q(x) and remainder R.
Calculates first derivative f'(x) and polynomial evaluation f(k).

Benefits of Using Polynomial Solver & Synthetic Division Calculator

Saves algebra and calculus students hours of manual division homework.

Provides instant verification for polynomial root factoring.

Practical Use Cases

Algebra & Calculus Students

Solve synthetic division and polynomial root factoring homework.

Math Teachers & Tutors

Generate verified step-by-step synthetic division answer keys.

Frequently Asked Questions

What is the difference between synthetic division and long division?

Synthetic division is a shorthand method of polynomial division that only uses coefficients instead of full variables, making it much faster when dividing by linear expressions of the form (x - k).

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