Multiplying Binomials Calculator
Expand two linear binomials, combine like terms, and inspect every FOIL contribution.
Example workbook validated and ready to download.
Binomial coefficients
Enter ordinary decimal numbers. Commas may be used only as thousands separators.
A First binomial: a₁x + a₀
Required. Example: 3 or – 0.5.
Required. Keep the minus sign for subtraction.
B Second binomial: b₁x + b₀
Required. Enter 1 when the binomial shows x.
Required. Example: 5, 0, or – 7.25.
Expanded result
Coefficient cards can preview valid pairs; the full product uses the complete input set.
Expanded product
3x² + 13x – 10
Step-by-step solution
(3x – 2)(x + 5) = (3 × 1)x² + (3 × 5)x + ( – 2 × 1)x + ( – 2 × 5) = 3x² + (15 – 2)x – 10 = 3x² + 13x – 10
The expanded product is 3x squared plus 13x minus 10.
FOIL contribution table
Each row comes from the same canonical calculation used by the result and Excel workbook.
| Contribution | Multiplication | Value | Feeds |
|---|---|---|---|
| First | 3 × 1 | 3 | c₂, the x² coefficient |
| Outer | 3 × 5 | 15 | One part of c₁ |
| Inner | – 2 × 1 | – 2 | One part of c₁ |
| Last | – 2 × 5 | – 10 | c₀, the constant |
| Combined middle | 15 + ( – 2) | 13 | c₁, the x coefficient |
FOIL is an organizational shortcut for applying the distributive property twice. The outer and inner products are like terms, so their values are added.
How to use the multiplying binomials calculator
What this calculator does
This calculator expands two linear binomials written in the form a₁x + a₀ and b₁x + b₀. It multiplies every term in the first binomial by every term in the second, combines the two like terms containing x, and returns the simplified polynomial c₂x² + c₁x + c₀. Use it to check algebra homework, verify a hand calculation, prepare a worked example, or inspect how a sign or coefficient changes the product. It is an exact algebraic identity for finite numeric coefficients; it does not solve an equation for x or factor a polynomial.
When to use it
Use the tool when expanding a product such as (3x – 2)(x + 5), checking the FOIL method, testing a special product such as conjugates, or confirming coefficients before substituting the polynomial into a larger problem. The underlying rule is the distributive property. OpenStax's guide to multiplying polynomials and the FOIL method gives additional examples and explains why all four term products must be included.
How to calculate
- The calculator opens with a complete demonstration: a₁ = 3, a₀ = – 2, b₁ = 1, and b₀ = 5. Its displayed result and validated Excel workbook are available immediately. Each coefficient must be finite and have an absolute value no greater than 10^154.
- Replace the four coefficients with your own values. Individual coefficient products can appear as soon as their required pairs are valid; the full expanded product appears after all four entries are valid. Use a standard period for decimals; commas are accepted only as thousands separators.
- Read the Expanded product first, then compare c₂, c₁, and c₀ with the FOIL contribution table. The table shows exactly where each coefficient comes from.
- Select Download Excel to export the current typed inputs, coefficients, expressions, FOIL rows, and notes as a real .xlsx workbook. Reset clears the demonstration and all calculated content; Download Excel then stays disabled until a complete valid set is entered again.
Input guide
x coefficient (a₁) is the required coefficient multiplying x in the first binomial. Enter a finite decimal such as 3, – 0.5, or 1,000. Increasing it changes both c₂ and the outer contribution to c₁. A common mistake is leaving it blank when the written term is simply x; enter 1 in that case, or – 1 for – x.
Constant (a₀) is the required constant term in the first binomial. Enter the sign explicitly, such as – 2 for “minus 2.” It changes the inner contribution to c₁ and the last product c₀. Do not type “x” or another unit in the field.
x coefficient (b₁) is the required coefficient multiplying x in the second binomial. A realistic example is 1 for x + 5. It affects c₂ and the inner contribution to c₁. Entering 0 is allowed and turns the second expression into a constant, so the final degree may drop below two.
Constant (b₀) is the required constant term in the second binomial. An example is 5. It controls the outer contribution to c₁ and the last product c₀. Decimal-comma input such as 1,5 is rejected because this calculator uses the period-decimal convention and treats commas only as correctly grouped thousands separators.
Output guide
Expanded product is the simplified polynomial identity. Quadratic coefficient c₂ equals a₁b₁, Linear coefficient c₁ equals a₁b₀ + a₀b₁, and Constant coefficient c₀ equals a₀b₀. The summary pills repeat Product degree, Linear coefficient, and Constant coefficient for fast scanning. A zero leading coefficient lowers the product degree; if all coefficients are zero, the result is the zero polynomial and its degree is undefined. Negative coefficient values are normal and indicate subtraction, not an error.
The Step-by-step solution displays the two original binomials, all four products, the combination of the outer and inner terms, and the final expression. In the FOIL contribution table, Contribution names the First, Outer, Inner, Last, or Combined middle step; Multiplication shows the exact coefficient operation; Value is that operation's numeric result; and Feeds identifies the resulting polynomial coefficient. These rows are exact identities, not estimates.
Worked example
With the startup values, the first product is 3 × 1 = 3, so c₂ = 3. The outer product is 3 × 5 = 15, while the inner product is – 2 × 1 = – 2. Combining those like terms gives c₁ = 15 + ( – 2) = 13. The last product is – 2 × 5 = – 10, so c₀ = – 10. Therefore, (3x – 2)(x + 5) = 3x² + 13x – 10. The first-open controls, result cards, table, step sequence, and workbook all use these same values.
Why the formula works
Multiplying binomials is repeated distribution. First distribute a₁x across the second binomial, then distribute a₀ across it. The two middle products both contain exactly one x, so they are like terms and can be added. OpenStax's College Algebra treatment of polynomial multiplication describes the general rule: multiply each term of one polynomial by each term of the other, then combine like terms.
= (a₁b₁)x² + (a₁b₀ + a₀b₁)x + a₀b₀
The FOIL mnemonic labels these products First, Outer, Inner, and Last. It is useful specifically for two binomials, while the distributive property works for polynomials of any length. For another concise walkthrough, see Khan Academy's multiplying binomials review.
Sign checks, special cases, and common mistakes
Most incorrect expansions come from losing a negative sign or omitting one of the two middle products. Keep each signed coefficient intact during multiplication. For example, ( – 2)(5) is – 10, while ( – 2)( – 5) is +10. The contribution table makes this audit explicit. Another useful check is substitution: choose a convenient value such as x = 1, evaluate the original product and the expanded polynomial, and confirm both totals match.
Some inputs produce a lower-degree result. If a₁ or b₁ is zero, then c₂ is zero and the product is linear or constant. Conjugate binomials such as (x + 4)(x – 4) produce opposite middle contributions that cancel, leaving x² – 16. Squared binomials such as (x + 4)² are entered by using the same coefficients for both binomials.