2026-09-29 8 min

Polynomial Expander (FOIL)

Expand (ax+b)(cx+d) into a quadratic with FOIL steps and numeric verification. Free online polynomial expander.

Calculator

Binomial Factors

a — first x-coefficient Help

Factor (ax + b)

b — first constant Help

Constant of (ax + b)

c — second x-coefficient Help

Factor (cx + d)

d — second constant Help

Constant of (cx + d)

Check value x Help

Verification point

Results update live as you type

Status:(2x + 3)(x − 4)= 2x² + -5x + -12

Expansion Summary

Factors(2x + 3)(x − 4)

F · O · I · L2 · -8 · 3 · -12

Expanded2x² + -5x + -12

Check at 513 = 13 ✓

Results

Optimal

Expanded Quadratic

2x² − 5x − 12

Standard form q₂x² + q₁x + q₀

x² Coefficient

First product a·c

x Coefficient

-5

Outer + Inner = ad + bc

Constant

-12

Last product b·d

Check at x = 5

13 = 13

Factored vs expanded evaluation must agree

Expansion uses exact FOIL arithmetic and is proved by evaluating both forms at your check x.

Visualizations

FOIL Product Contributions

2First

8Outer

3Inner

12Last

Verification

Match ✓100%

Factored vs Expanded

Factored13

Formula

Each term of the first binomial multiplies each term of the second; the two x-products merge.

(ax + b)(cx + d) = ac·x² + (ad + bc)·x + bd

Alternative Form

FOIL: First + Outer + Inner + Last

Variables

Symbol	Name	Unit	Description
a, b	First binomial	numbers	Coefficients of (ax + b)
c, d	Second binomial	numbers	Coefficients of (cx + d)
q₁	Middle coefficient	number	Merged Outer + Inner = ad + bc
x	Check point	number	Value where both forms are compared

Explanation

The four FOIL tiles show where each product lands; the two middle tiles stack into the single x-term.

Derivation

Distribute twice: (ax+b)(cx+d) = ax·cx + ax·d + b·cx + b·d = acx² + (ad+bc)x + bd.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Write the two factors.

(ax + b)(cx + d)

F1:(2x + 3)

F2:(x − 4)

Result: (2x + 3)(x − 4)

Step 2

Multiply First terms.

F = a·c·x²

F:2×1 = 2

Result: F = 2x²

Step 3

Multiply Outer and Inner terms.

O = a·d·x, I = b·c·x

O:2×-4 = -8

I:3×1 = 3

Result: O + I = -8x + 3x

Step 4

Multiply Last terms.

L = b·d

L:3×-4 = -12

Result: L = -12

Step 5

Merge the middle terms into one x coefficient.

q₁ = ad + bc

q1:-8 + 3 = -5

Result: Expanded: 2x² + -5x + -12

Step 6

Verify at the check x.

LHS vs RHS

x:5

LHS:13

RHS:13

Result: 13 = 13 ✓

Final Answer

2x² − 5x − 12

Worked Examples

See how the calculator works with different scenarios.

Scenario

Standard FOIL exercise.

Inputs

a:2

b:3

c:1

d:-4

x:5

Calculation

F=2x², O=−8x, I=3x, L=−12 → 2x²−5x−12

Result

2x² − 5x − 12

Interpretation

Check at 5: 13×1 = 13 and 50−25−12 = 13 ✓.

Understanding Your Result

Your Result: Expanded: 2x² + -5x + -12

Typical range: Any real a, b, c, d; q₂ = 0 collapses the result to linear (degenerate binomial product).

What This Means

The quadratic lists exactly what each power contributes after every pair is multiplied out.

Why It Matters

Expanding unlocks the quadratic formula, vertex form, graphing and area models — factored form hides all of that.

Professional Interpretation

Read q₁ = ad + bc as the cross-coupling of the factors; symmetric factors give the 2ab perfect-square pattern.

Recommendations

•Always merge O + I before finishing
•Verify with one numeric check
•Watch signs on negative constants
•Spot squares and differences early
•Expand pairwise for longer products

Educational Content

Introduction

Expansion turns compact factored form into explicit standard form, exposing every coefficient FOIL creates.

Concept Explanation

Double distribution: each of 2 terms times each of 2 terms gives 4 products; equal powers add.

Applications

▸Preparing quadratics for the formula

▸Area and product models

▸Completing the square setup

▸Derivative of factored forms

▸Checking factored answers

▸Physics product relations

Benefits

•Four products itemized
•Middle merge shown
•Automatic numeric proof
•Sign-safe arithmetic
•Pattern spotting (squares)

Limitations

•Binomial × binomial only
•Numeric coefficients
•Single variable

Best Practices

•Write factors before multiplying
•Box or grid the four products
•Merge middles explicitly
•Verify at an easy x like 1
•Keep sign with each constant

Common Misconceptions

Myth: (a + b)² = a² + b².

Truth: The middle 2ab survives: (x+3)² = x² + 6x + 9, not x² + 9.

Myth: FOIL works for any product.

Truth: FOIL is binomial×binomial only; larger products need full distribution.

Myth: Outer and Inner stay separate.

Truth: Both are x-terms and must merge into one coefficient.

Myth: Signs can be fixed at the end.

Truth: Attach each sign before multiplying, or L and middles flip.

Practical Tips

•Attach signs before multiplying
•Always merge Outer + Inner
•Verify at x = 1 for a fast check
•Spot (a±b)² and difference patterns
•Draw the 2×2 box when learning
•Expand pairwise for 3+ factors
•Re-check L = b·d separately

Common Mistakes

Forgetting the middle term (a+b)² → a²+b²

Add 2ab: Outer + Inner never vanish except in difference patterns.

Sign errors on negative constants

Keep the minus with the number through every product.

Leaving O and I unmerged

Both multiply x — add them into a single coefficient.

Multiplying only First and Last

All four pairs count; missing O/I loses the x-term.

Skipping verification

One check-x comparison catches nearly every arithmetic slip.

Assumptions

This calculator makes the following assumptions to provide estimates:

Binomial factors: Each factor is exactly degree 1 in x.

Real coefficients: a, b, c, d and x are real numbers.

Exact arithmetic: Inputs are exact, not rounded measurements.

Single variable: Both factors share the variable x.

Display rounding: Results show up to 6 decimals.

Limitations

Be aware of these limitations when using this calculator:

Two binomials only: Trinomials need full distribution, not FOIL.

Numeric scope: Symbolic parameters (like k) are unsupported.

One variable: Multivariate products are out of scope.

Degenerate case: Zero x-coefficients collapse the quadratic to linear.

Frequently Asked Questions

Answers to common questions about this calculator.

What does expanding mean?

Expanding removes parentheses by multiplying every term pair: (ax + b)(cx + d) becomes ac·x² + (ad + bc)·x + bd. It is the reverse of factoring.

What is the FOIL method?

FOIL = First, Outer, Inner, Last: the four products of two binomials. For (2x + 3)(x − 4): F = 2x², O = −8x, I = 3x, L = −12, merging to 2x² − 5x − 12.

How do I use this calculator?

Enter a and b for the first factor (ax + b), c and d for the second (cx + d), plus a check value x. The quadratic coefficients and FOIL steps appear instantly.

Do the middle terms always combine?

Yes — Outer and Inner are both x-terms, so ad·x + bc·x = (ad + bc)·x. Forgetting to merge them is the most common FOIL error.

How do I verify the expansion?

Pick any x and evaluate both sides. This page checks automatically: (ax + b)(cx + d) must equal q₂x² + q₁x + q₀ at your check x.

What about squared binomials like (x + 3)²?

Set both factors equal: a = c = 1, b = d = 3 gives x² + 6x + 9 — the perfect-square pattern (a + b)² = a² + 2ab + b².

What is the difference of squares?

(x + b)(x − b) = x² − b²: the middle terms cancel. Try a = c = 1, b = 4, d = −4 to see O + I = 0.

Can I expand larger products?

This page covers binomial × binomial. For bigger products, expand pairwise: multiply two factors, then multiply the result by the next.

Related Calculators

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Related Guides

Deepen your understanding with these educational resources.

FOIL without sign errors

Box method walkthrough.

Special products cheat sheet

Squares and differences.

From expanded to vertex

Next step after expanding.

Checking by substitution

One-x verification habit.

Glossary

Technical terms used in mortgage calculations explained in plain language.

FOIL: First, Outer, Inner, Last — the four binomial products.
Binomial: Two-term expression ax + b.
Expanded form: Standard polynomial with parentheses removed.
Factored form: Polynomial written as a product of factors.
Middle term: Merged Outer + Inner x-coefficient.
Perfect square: (a ± b)² = a² ± 2ab + b².
Difference of squares: (a+b)(a−b) = a² − b².
Distribution: Multiplying each term across a sum.

Authoritative References

Trusted sources and standards referenced by this calculator.

Khan Academy — Multiplying binomials Math is Fun — FOIL Purplemath — Multiplying polynomials Wikipedia — FOIL method Wolfram MathWorld — FOIL Cuemath — FOIL method

Applicable Standards

Common Core A-APR — polynomial arithmeticNCTM Algebra — symbolic manipulation

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Page Information

Last Updated:

2026-09-29

Version:

2.0.0

References:

License:

Free to use for educational and professional purposes.