Multiplying Binomials Calculator

Multiply two binomials (ax + b)(cx + d) using the FOIL method with step-by-step solutions.

Enter Binomial Coefficients

First Binomial (ax + b)
Second Binomial (cx + d)
(2x + 3)(4x + 5)

Result

Product of Binomials
8x² + 22x + 15
(2x + 3)(4x + 5)
First (F): ax × cx 8x²
Outer (O): ax × d 10x
Inner (I): b × cx 12x
Last (L): b × d 15
x² coefficient 8
x coefficient (combined) 22
Constant term 15

Step-by-Step FOIL Solution

(ax+b)(cx+d) = acx² + (ad+bc)x + bd

What Is the FOIL Method?

FOIL is a mnemonic that stands for First, Outer, Inner, Last. It is a technique for multiplying two binomials, which are algebraic expressions containing exactly two terms each. The FOIL method systematically ensures that every term in the first binomial is multiplied by every term in the second binomial.

The FOIL Steps Explained

F - First Terms

Multiply the first terms of each binomial together.

(ax)(cx) = acx²

O - Outer Terms

Multiply the outermost terms in the product.

(ax)(d) = adx

I - Inner Terms

Multiply the two innermost terms.

(b)(cx) = bcx

L - Last Terms

Multiply the last terms of each binomial.

(b)(d) = bd

The General Formula

When you multiply (ax + b)(cx + d), the FOIL method gives you:

(ax + b)(cx + d) = acx² + (ad + bc)x + bd

The result is always a trinomial (three terms) unless like terms cancel. The x² coefficient comes from the First terms, the x coefficient comes from combining Outer and Inner terms, and the constant comes from the Last terms.

Special Products

  • Perfect Square Trinomial: (ax + b)² = a²x² + 2abx + b²
  • Difference of Squares: (ax + b)(ax - b) = a²x² - b² (the middle term cancels)
  • Sum and Product Pattern: (x + p)(x + q) = x² + (p+q)x + pq

Why FOIL Matters

The FOIL method is one of the most fundamental algebraic techniques taught in algebra courses. It is the foundation for:

  • Expanding polynomial expressions
  • Factoring quadratic equations (reverse FOIL)
  • Completing the square
  • Understanding the quadratic formula
  • Working with higher-degree polynomials

Beyond FOIL

While FOIL works specifically for multiplying two binomials, the underlying principle is the distributive property of multiplication over addition. For multiplying polynomials with more than two terms, you would use the general distributive property (sometimes called the "claw" or "grid" method) instead of FOIL.