Multiply Polynomials Calculator
Multiply polynomials instantly using the FOIL method for binomials or the distribution method for any polynomial — see every cross product, the like terms collection step, and automatic special product detection.
Enter two binomials to multiply polynomials using FOIL (First, Outer, Inner, Last). Format: (3x+2)(x-5)
FOIL Arc Diagram — First, Outer, Inner, Last
Result of Multiplying Polynomials (FOIL)
This tool will multiply polynomials of any size using full distribution — every term times every term, then like terms collected.
Distribution Grid — Every Term × Every Term
Vertical Multiplication Layout
Final Result — Multiply Polynomials
Distribute a monomial across a polynomial: a(b+c) = ab+ac.
Result
Click any special product below to load a live example into the FOIL or Multiply Polynomials tool.
Multiply Polynomials with FOIL or Distribution — Complete Guide
This multiply polynomials calculator handles every case: use the FOIL method for two binomials, or the general distribution method for polynomials of any size. Every cross product is shown, like terms are collected step by step, and special products such as the difference of squares and perfect square trinomials are detected automatically. Whether you need to multiply polynomials for homework, multiply two binomials for a quiz, or check a distribution-method expansion, this multiply polynomials calculator shows the full working.
The FOIL Method — Multiplying Two Binomials
FOIL stands for First, Outer, Inner, Last — a mnemonic for the four multiplications required when you multiply polynomials that are both binomials. The FOIL method only works for binomial × binomial (two terms times two terms). For anything larger, use the general distribution method described below.
Given (ax+b)(cx+d), the FOIL method identifies:
- First: multiply the first term of each binomial — (ax)(cx) = acx²
- Outer: multiply the outermost terms — (ax)(d) = adx
- Inner: multiply the innermost terms — (b)(cx) = bcx
- Last: multiply the last term of each binomial — (b)(d) = bd
After the FOIL method produces four products, the Outer and Inner terms are like terms (both degree 1) and must be combined: ad+bc. The FOIL method is really just the distribution method applied systematically to a binomial × binomial case — every term of the first binomial is distributed across every term of the second.
Worked Example 1: (x+3)(x+2)
F: x·x = x² O: x·2 = 2x I: 3·x = 3x L: 3·2 = 6
Combine O+I: 2x+3x = 5x
Result: x² + 5x + 6
Worked Example 2: (2x−1)(3x+4)
F: 2x·3x = 6x² O: 2x·4 = 8x I: −1·3x = −3x L: −1·4 = −4
Combine O+I: 8x−3x = 5x
Result: 6x² + 5x − 4
Worked Example 3: (x+5)(x−5) — Special Product
F: x·x = x² O: x·(−5) = −5x I: 5·x = 5x L: 5·(−5) = −25
Combine O+I: −5x+5x = 0 (they cancel!)
Result: x² − 25 — this is the difference of squares special product.
Whenever you multiply polynomials using FOIL and the Outer/Inner terms cancel to zero, you've found a difference of squares. When they double instead of canceling, you've found a perfect square trinomial.
How to Multiply Polynomials — The Distribution Method
To multiply polynomials with more than two terms, use full distribution: multiply every term of the first polynomial by every term of the second polynomial, then collect like terms. If polynomial P has m terms and polynomial Q has n terms, distribution produces exactly m×n cross products before combining like terms. FOIL is simply the m=2, n=2 special case of this same distribution method.
Example A — Monomial × Binomial
3x(x+4) = 3x·x + 3x·4 = 3x² + 12x
Example B — Binomial × Trinomial
(x+2)(x²−x+3): distribute x and 2 across all three terms.
x·x²=x³, x·(−x)=−x², x·3=3x, 2·x²=2x², 2·(−x)=−2x, 2·3=6
Like terms: x² terms −x²+2x²=x²; x terms 3x−2x=x
Result: x³ + x² + x + 6
Example C — Trinomial × Trinomial
(2x²+x−3)(x²−2x+1) has 3×3=9 cross products before like terms collection. Use the calculator above to see the full distribution grid and every one of the 9 products.
Special Products — Shortcuts Worth Memorizing
Special products are polynomial multiplications whose results follow a fixed pattern, letting you skip full FOIL or distribution. All five special products below can be verified using FOIL or distribution — they are simply the pre-simplified results.
These are special because the cross terms either cancel completely (difference of squares — Outer and Inner are opposites) or combine identically (perfect square trinomial — Outer and Inner are equal, so they double instead of canceling). Recognizing these patterns saves significant computation time on exams whenever you multiply polynomials that fit the pattern.
Multiplying Polynomials Step by Step — The Grid Method
The grid (box) method is a visual alternative to FOIL or distribution. Draw a table with the terms of P as row headers and the terms of Q as column headers. Fill each cell with the product of its row and column term, then collect like terms by summing cells that share the same resulting degree. The grid method for multiplying polynomials is especially useful for larger polynomials because it guarantees you never miss a cross product — the multiply polynomials calculator above generates this grid automatically for any two polynomials you enter.
| × | x² | −4x | +3 |
|---|---|---|---|
| 3x² | 3x⁴ | −12x³ | 9x² |
| +2x | 2x³ | −8x² | 6x |
| −1 | −x² | +4x | −3 |
Reading diagonals of like degree: x⁴ term = 3x⁴; x³ terms = −12x³+2x³=−10x³; x² terms = 9x²−8x²−x²=0; x¹ terms = 6x+4x=10x; constant = −3. Result: 3x⁴−10x³+10x−3.
Multiplying Monomials and Polynomials — Distribution
When the first factor is a single monomial, multiply that monomial by each term of the polynomial separately using the distributive property: a(b+c) = ab+ac. For the variable part, add exponents (product rule for exponents: xᵃ·xᵇ=xᵃ⁺ᵇ).
3x²(2x³−x+4): 3x²·2x³=6x⁵, 3x²·(−x)=−3x³, 3x²·4=12x². Result: 6x⁵−3x³+12x²
−2x(x²+3x−1): −2x·x²=−2x³, −2x·3x=−6x², −2x·(−1)=2x. Result: −2x³−6x²+2x
5(x²−4x+3): 5·x²=5x², 5·(−4x)=−20x, 5·3=15. Result: 5x²−20x+15
Common Mistakes When Multiplying Polynomials
- Using FOIL on non-binomials: FOIL gives exactly 4 products for a binomial × binomial. A trinomial × trinomial has 9 cross products — FOIL cannot be applied; use full distribution instead.
- Sign errors: (x−3)(x−2) gives a Last term of (−3)(−2)=+6, not −6. Negative × negative = positive.
- Adding exponents incorrectly: x²×x³ = x⁵ (add exponents), not x⁶ (never multiply exponents when multiplying like bases).
- Missing cross products: forgetting to multiply every term by every term — always count m×n products before combining like terms.
- Combining unlike terms: 3x² and 3x are not like terms (different degrees) and cannot be added together.
Worked Examples — 10 Complete Problems
1. (x+4)(x+3) = x²+7x+12
2. (x−2)(x+5) = x²+3x−10
3. (2x+3)(4x−1) = 8x²+10x−3
4. (x+7)(x−7) = x²−49 (difference of squares)
5. (3x−4)² = 9x²−24x+16 (perfect square trinomial)
6. x²(3x³−2x+5) = 3x⁵−2x³+5x²
7. (x+2)(x²−x+3) = x³+x²+x+6
8. (x²−1)(x²+x+1) = x⁴+x³−x−1
9. (2x²+x−3)(x²−2x+1) — full trinomial × trinomial (9 cross products)
10. (x+1)³ = (x+1)²×(x+1) = x³+3x²+3x+1