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Polynomial Multiplication Calculator

Multiply two linear binomials (a₁x + a₀) and (b₁x + b₀) using the FOIL method. Enter four coefficients and read the quadratic result term by term.

First polynomial: a1·x + a0

Second polynomial: b1·x + b0

Result

2x^2 - 5x - 12

(2x + 3)(1x - 4) = 2x^2 - 5x - 12

Quick Answer

A polynomial multiplication calculator applies FOIL to (a₁x + a₀)(b₁x + b₀): the x² coefficient is a₁b₁, the x coefficient is a₁b₀ + a₀b₁, and the constant is a₀b₀. For (2x + 3)(4x + 5): x² term = 2 × 4 = 8, x term = 2 × 5 + 3 × 4 = 22, constant = 3 × 5 = 15. Result: 8x² + 22x + 15.

Two Linear Binomials: What This Calculator Multiplies

This calculator multiplies exactly two linear binomials of the form (a₁x + a₀) and (b₁x + b₀), where each binomial has a degree-1 (x) term and a constant term. The result is always a quadratic trinomial of the form Ax² + Bx + C. This is the FOIL case: First, Outer, Inner, Last. It does not multiply higher-degree polynomials such as quadratics, cubics, or polynomials with more than two terms.

The FOIL Formula for Two Linear Binomials

(a₁x + a₀)(b₁x + b₀) = a₁b₁x² + (a₁b₀ + a₀b₁)x + a₀b₀

  • (a₁x + a₀)(b₁x + b₀) = a₁b₁x² + (a₁b₀ + a₀b₁)x + a₀b₀
  • First: a₁x × b₁x = a₁b₁x²
  • Outer: a₁x × b₀ = a₁b₀x
  • Inner: a₀ × b₁x = a₀b₁x
  • Last: a₀ × b₀ = a₀b₀

How To Use the Polynomial Multiplication Calculator

Inputs

  • a₁: coefficient of x in the first binomial
  • a₀: constant in the first binomial
  • b₁: coefficient of x in the second binomial
  • b₀: constant in the second binomial

Steps

  1. Identify the x-coefficient and constant of each binomial.
  2. Enter a₁ and a₀ for the first binomial.
  3. Enter b₁ and b₀ for the second binomial.
  4. Read the three coefficients of the result: A (for x²), B (for x), and C (the constant).
  5. Write the result as Ax² + Bx + C.

FOIL Worked Example: (3x − 2)(5x + 4)

Multiplying (3x − 2) by (5x + 4) using the FOIL expansion method.

  1. First (x² term): A = 3 × 5 = 15.
  2. Outer + Inner (x term): B = (3)(4) + (−2)(5) = 12 − 10 = 2.
  3. Last (constant term): C = (−2)(4) = −8.

(3x − 2)(5x + 4) = 15x² + 2x − 8.

When FOIL Applies and When a Different Method Is Needed

FOIL applies specifically and only to the product of two linear binomials. Use this calculator when both factors are of degree 1 with exactly two terms each. For multiplying higher-degree polynomials or trinomials, use full term-by-term distribution.

Assumptions

  • Both factors are linear binomials of the form (a₁x + a₀).
  • Coefficients may be any real numbers, including negatives and fractions.
  • The result is a quadratic; if a₁ or b₁ equals 0, the result reduces to a linear expression.

Limitations

  • Does not multiply polynomials of degree 2 or higher.
  • Does not handle factors with more than two terms.
  • Does not factor or simplify the result beyond expanding.

In Practice

The most common FOIL errors are sign mistakes with negative constants and forgetting to add the Outer and Inner terms before writing the x coefficient. When one constant is negative (e.g., 3x − 2), the Last term (a₀ × b₀) is negative and the Inner term (a₀b₁x) is also negative. Write out all four products explicitly before combining to avoid dropped signs.

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Frequently Asked Questions: Polynomial Multiplication Calculator

What does FOIL stand for?

FOIL stands for First, Outer, Inner, Last: the four pairs of terms you multiply when expanding the product of two binomials. First: the leading terms. Outer: first term of the first binomial and last term of the second. Inner: last term of the first binomial and first term of the second. Last: the constant terms.

Can FOIL handle polynomials with more than two terms?

No. FOIL is a memory aid specific to two-binomial (two-term × two-term) multiplication. For a binomial times a trinomial, or any product of polynomials with three or more terms, use full distribution: multiply each term of the first polynomial by every term of the second, then collect like terms.

What happens when a₁ or b₁ is zero?

If a₁ = 0, the first binomial reduces to the constant a₀, and the product is simply a₀(b₁x + b₀) = a₀b₁x + a₀b₀, a linear result. The calculator will still compute the correct answer but the x² coefficient will be 0.

What if the result has a zero x term (B = 0)?

This happens when a₁b₀ = −a₀b₁, meaning the Outer and Inner terms cancel. The result is a difference of squares: (ax + b)(ax − b) = a²x² − b², where the x term vanishes.

Can I use this for complex or fractional coefficients?

Yes. The formula works for any real (or complex) number coefficients. Enter fractions as decimals or exact fractions and carry the arithmetic through the four FOIL products manually to verify.

Sources

Last updated: . Reviewed for accuracy against the formula shown above.