Multiplying Binomials Calculator
Enter the coefficients and constants for two binomials of the form (a1x + a0)(b1x + b0). The calculator expands the product using the FOIL method, shows every step, identifies the special pattern if one applies, and displays the simplified trinomial result.
Formula
Worked example
Multiply (3x - 2)(x + 5): F = 3x * x = 3x², O = 3x * 5 = 15x, I = -2 * x = -2x, L = -2 * 5 = -10. Combine: 3x² + (15x - 2x) - 10 = 3x² + 13x - 10.
What is the FOIL method?
FOIL stands for First, Outer, Inner, Last - a mnemonic for multiplying two binomials by applying the distributive property twice. "First" means multiply the leading terms of each binomial together. "Outer" means multiply the first term of the first binomial by the last term of the second. "Inner" means multiply the last term of the first binomial by the first term of the second. "Last" means multiply the trailing constants together. After you have those four products, combine any like terms (the Outer and Inner terms both have a single x, so they add together) to get the final trinomial in standard form: ax² + bx + c.
Special product patterns to recognize
Two patterns occur so often that they are worth memorizing. A Perfect Square Trinomial appears when you square a binomial: (ax + b)² = a²x² + 2abx + b². You can spot it when both binomials are identical - the middle coefficient always equals twice the product of the two term multipliers. The Difference of Squares pattern appears when the two binomials are conjugates: (ax + b)(ax - b) = a²x² - b². The outer and inner products cancel each other exactly, leaving no linear term. Recognizing these patterns lets you skip the full FOIL steps and write the answer directly, which is especially useful when factoring trinomials in reverse.
How to multiply binomials step by step
Start by labeling your binomials (a1x + a0) and (b1x + b0). Compute the four products: a1 times b1 (x² term), a1 times b0 (an x term), a0 times b1 (another x term), and a0 times b0 (constant). Add the two x terms together. Write the result as c2x² + c1x + c0 where c2 = a1*b1, c1 = a1*b0 + a0*b1, and c0 = a0*b0. Always check the signs carefully - a negative constant in a binomial means you are subtracting, not adding, so the sign carries through into the product.
Why multiplying binomials matters in algebra
Multiplying binomials is the direct inverse of factoring quadratics, which is one of the most common tasks in algebra, precalculus, and calculus. Every time you factor a trinomial ax² + bx + c you are finding two binomials whose product gives that trinomial back. Understanding the FOIL structure also underpins completing the square, the quadratic formula derivation, and polynomial long division. In higher math, the same distributive logic extends to multiplying polynomials of any degree using the same term-by-term principle.
Special Binomial Product Patterns
| Pattern name | Form | Result |
|---|---|---|
| Perfect Square Trinomial (sum) | (ax + b)² | a²x² + 2abx + b² |
| Perfect Square Trinomial (diff) | (ax - b)² | a²x² - 2abx + b² |
| Difference of Squares | (ax + b)(ax - b) | a²x² - b² |
| Sum of Binomials | (x + a)(x + b) | x² + (a+b)x + ab |
| General FOIL | (a1x + a0)(b1x + b0) | a1b1x² + (a1b0+a0b1)x + a0b0 |
Memorizing these patterns speeds up both multiplication and factoring.
Frequently asked questions
What does FOIL stand for?
FOIL stands for First, Outer, Inner, Last. It describes the four pairs of terms you multiply when expanding the product of two binomials: First refers to the leading x terms of each, Outer to the first term of the left binomial and the constant of the right, Inner to the constant of the left binomial and the x term of the right, and Last to the two constants. Combining the four products and simplifying gives the expanded trinomial.
Can I use FOIL for more than two terms?
FOIL only applies to two binomials (two terms each). For larger polynomials you use the general distributive property: multiply every term in the first polynomial by every term in the second, then collect like terms. FOIL is simply the four-step shortcut that works when each factor has exactly two terms.
What is a difference of squares?
A difference of squares is the product of two conjugate binomials: (ax + b)(ax - b). Because the outer product is +abx and the inner product is -abx, they cancel, leaving only a²x² - b². This pattern shows up frequently in factoring and simplifying rational expressions.
What is a perfect square trinomial?
A perfect square trinomial results from squaring a binomial: (ax + b)² = a²x² + 2abx + b². The key feature is that the middle coefficient equals twice the product of the two outer coefficients, and the constant term is a perfect square. Recognizing this pattern makes factoring the trinomial back into (ax + b)² quick and straightforward.
Does the order of the binomials matter?
No. Multiplication is commutative, so (ax + b)(cx + d) equals (cx + d)(ax + b). The FOIL labels First and Last swap, and Outer and Inner swap, but the four products are the same. The final expanded trinomial is identical regardless of which binomial you call first.