expand calculator

What this expand calculator does

This tool expands two common algebra forms:

• Product of two binomials: (ax + b)(cx + d)
• Power of a binomial: (ax + b)n

Instead of manually distributing terms and combining coefficients, you can enter values and get the expanded polynomial in standard form. It works with positive numbers, negative numbers, and decimals.

How expansion works

1) Expanding (ax + b)(cx + d)

The distributive rule gives:

(ax + b)(cx + d) = acx2 + (ad + bc)x + bd

So the final quadratic has:

• Leading coefficient: ac
• Middle coefficient: ad + bc
• Constant: bd

2) Expanding (ax + b)n

This uses the binomial theorem:

(ax+b)n = Σ [C(n,k) · (ax)n-k · bk], for k = 0 ... n

Each term is computed with a binomial coefficient and placed into descending powers of x. The calculator handles this automatically and outputs a clean polynomial.

Why this is useful

• Check homework or exam practice quickly.
• Verify symbolic simplifications in engineering and physics calculations.
• Build intuition around coefficient patterns and Pascal triangle behavior.
• Save time when testing many coefficient combinations.

Tips for best results

Use exact values when possible

If you use decimals, tiny floating-point rounding can appear in long expansions. The calculator rounds output to keep expressions readable.

Keep powers practical

For (ax+b)n, large values of n produce very large coefficients. This page supports 0 ≤ n ≤ 20, which is a practical range for most users.

Interpret signs carefully

Negative inputs are fully supported. For example, b = -4 means your binomial is ax - 4, which changes multiple terms during expansion.

Quick examples

• (2x + 3)(4x - 5) = 8x2 + 2x - 15
• (x + 2)3 = x3 + 6x2 + 12x + 8
• (3x - 1)2 = 9x2 - 6x + 1

Final thought

A good expand calculator is more than a shortcut—it helps you learn structure. Use the computed polynomial, then compare it against your own manual expansion to strengthen algebra fluency.