2026-09-29 9 min

Polynomial Multiplication Calculator

Multiply two polynomials by convolution with grid steps, degree tracking and triple verification. Free online multiplier.

Calculator

Factors

P coefficients

First factor, highest-degree-first.

Q coefficients

Results update live as you type

Status:(x + 2) × (x + 3)= x^2 + 5x + 6

Multiplication Summary

Grid size2 × 2 = 4 pairs

Productx^2 + 5x + 6

Degree2

Pair grid[1, 2] [3, 6]

Results

Optimal

Product P·Q

x^2 + 5x + 6

Convolution of the coefficient lists

Degree

1 + 1 = 2

Degrees add under multiplication

Non-Zero Terms

Surviving diagonals

Leading Check

1 = 1

lead(P)·lead(Q) must open the product

Check P(1)·Q(1)

3×4 = 12

Evaluation homomorphism check

Convolution arithmetic is exact and triple-verified by leading, constant and evaluation checks.

Visualizations

Product Coefficients by Power

1x^2

5x^1

6x^0

Convolution Complete

2×2 pairs100%

|P(1)| vs |Product(1)|

|P(1)|3

Formula

Each output slot collects every pair whose positions sum to k — diagonals of the product grid.

prod_k = Σ pᵢ·qⱼ over all i + j = k (convolution)

Alternative Form

deg(P·Q) = deg P + deg Q

Variables

Symbol	Name	Unit	Description
P	First factor	coeffs	Highest-degree-first list
Q	Second factor	coeffs	Highest-degree-first list
prod_k	Diagonal sum	number	Total of grid diagonal k
deg	Product degree	integer	Sum of input degrees

Explanation

Grid cells are pair products; each output bar stacks one diagonal of that grid.

Derivation

(Σ pᵢxⁱ)(Σ qⱼxʲ) = Σₖ(Σᵢ₊ⱼ₌ₖ pᵢqⱼ)xᵏ — collecting equal powers gives the convolution sum.

Step-by-Step Calculation

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

Step 1

Parse both factors positionally.

P and Q highest-first

P:x + 2

Q:x + 3

Result: (x + 2) × (x + 3)

Step 2

Build the pair-product grid: every P term times every Q term.

cell(i,j) = pᵢ·qⱼ

grid:2×2 = 4 pairs

Intermediate: 1, 2 | 3, 6

Result: 4 pair products formed

Step 3

Sum each diagonal into one product coefficient.

prod_k = Σ pᵢqⱼ over i+j=k

diagonals:x^2: 1 · x^1: 5 · x^0: 6

Intermediate: x^2: 1 | x^1: 5 | x^0: 6

Result: Slots: 1, 5, 6

Step 4

Confirm the degree rule and leading product.

deg = m+n; lead = lead(P)·lead(Q)

degree:2

lead:1

Result: Degree 2, opens with 1

Step 5

Triple-verify: constant, leading and x = 1 checks.

P(1)·Q(1) = S(1)

const:6

eval:12

Result: All three checks pass ✓

Final Answer

x^2 + 5x + 6

Worked Examples

See how the calculator works with different scenarios.

Scenario

FOIL as 2×2 convolution.

Inputs

p:1,2

q:1,3

Calculation

Grid 1,3,2,6 → diagonals 1, 5, 6

Result

x² + 5x + 6

Interpretation

Middle diagonal 3+2 = 5 is FOIL O+I.

Understanding Your Result

Your Result: P·Q = x^2 + 5x + 6

Typical range: Degree = sum of degrees; term count up to (m+1)(n+1) diagonals merged.

What This Means

The product blends every scale interaction between the factors — area, joint probability, filtered signals.

Why It Matters

Multiplication builds higher models from simple pieces and inverts factoring: expand to check any split.

Professional Interpretation

Read diagonals, not cells: each output coefficient is one diagonal sum — that is the whole algorithm.

Recommendations

•Draw the grid for 3+ term factors
•Check lead × lead first
•Check const × const second
•Evaluate at 1 as final proof
•Watch middle pile-ups

Educational Content

Introduction

Multiplication generalizes FOIL: the pair grid grows to any size, and diagonals become the coefficients.

Concept Explanation

Distribute every term across every term, then collect equal powers — convolution in one dimension.

Applications

▸Area/volume expansion

▸Probability generating functions

▸Digital filter convolution

▸Taylor series products

▸Characteristic polynomials

▸Checking factorizations

Benefits

•Full pair grid exposed
•Diagonal sums itemized
•Degree rule confirmed
•Triple verification
•Sign-safe cells

Limitations

•Two factors per run
•Numeric coefficients
•Single variable

Best Practices

•Tabulate the grid first
•Sum diagonals carefully
•Verify lead and constant
•Evaluate at x = 1
•Multiply pairwise for 3+ factors

Common Misconceptions

Myth: Multiply matching positions only.

Truth: Every P term hits every Q term — the grid is full, not diagonal.

Myth: Degrees multiply.

Truth: Degrees add: quadratic × cubic is quintic, not degree 6.

Myth: FOIL covers all products.

Truth: FOIL is the 2×2 special case; bigger factors need the full grid.

Myth: Middle terms stay separate.

Truth: Each diagonal merges into exactly one coefficient.

Practical Tips

•Build the full pair grid
•Sum diagonals into slots
•Verify lead × lead
•Verify const × const
•Evaluate at x = 1 as proof
•Remember degrees add
•Multiply pairwise for longer chains

Common Mistakes

Only multiplying aligned positions

Form all m×n pairs; alignment applies to sums, not products.

Forgetting to merge diagonals

Each diagonal collapses to one coefficient — add them.

Adding degrees instead of multiplying setup

Degree rule: output degree = m + n.

Sign slips in grid cells

Attach each factor sign before multiplying the cell.

Skipping verification

Lead, constant and x=1 checks catch nearly every slip.

Assumptions

This calculator makes the following assumptions to provide estimates:

Positional lists: First number is each highest-power coefficient.

Same variable: Both factors share x.

Real numbers: All coefficients are real.

Exact values: Inputs are exact, not measured.

Display rounding: Outputs round to 6 decimals.

Limitations

Be aware of these limitations when using this calculator:

Two factors: Chain longer products through the output.

Numeric scope: Symbolic factors unsupported.

Single variable: Multivariate products out of scope.

Grid display: Large grids summarize as diagonal sums.

Frequently Asked Questions

Answers to common questions about this calculator.

How do I multiply two polynomials?

Multiply every term of P by every term of Q and add matching powers — discrete convolution. (x + 2)(x + 3) = x² + 5x + 6. Enter both coefficient lists and the grid diagonal sums appear.

How do I enter the factors?

As comma-separated coefficients highest-degree-first, e.g. P = "1,2" for x + 2 and Q = "1,3" for x + 3. Interior zeros hold missing powers.

What is the convolution view?

Write P across the top and Q down the side of a grid; each cell is one pair product. Product coefficient k sums diagonal k — exactly how this calculator builds the answer.

What happens to the degrees?

They add: deg(P·Q) = deg P + deg Q. Quadratic × cubic is always quintic (unless a leading product cancels, which only zero factors cause).

How do I verify a product?

Two instant checks: the leading product must equal lead(P)·lead(Q), and the constant must equal const(P)·const(Q). A third: evaluate at x = 1 — P(1)·Q(1) must equal product(1).

Why do middle terms pile up?

Several diagonals land on the same power: in (x²+x+1)(x²+x+1) the x² slot collects three pair products. FOIL shows this with just two diagonals; convolution generalizes it.

What is special about (a+b)(a−b)?

Middle diagonals cancel to zero, leaving a² − b². The calculator shows the zeroed middle bar explicitly — the difference-of-squares signature.

Where is polynomial multiplication used?

Area and volume models, probability (generating functions), signal filtering (convolution), Taylor products, and building characteristic equations.