Multiply two binomials using the FOIL method (First, Outer, Inner, Last). See step-by-step solutions showing how to expand (ax + b)(cx + d).
Product
x^2 + 5x + 6
(x + 3) * (x + 2)
Enter coefficients for (ax + b)(cx + d)
(x + 3)
(x + 2)
Multiplying:
(x + 3) * (x + 2)
First
1x * 1x
1x^2
Outer
1x * 2
2x
Inner
3 * 1x
3x
Last
3 * 2
6
Expanded (before combining):
1x^2 +2x +3x +6
Simplified (final answer):
x^2 + 5x + 6
x^2 coefficient
1
x coefficient
5
constant
6
First
1 * 1 = 1
1x^2
Outer
1 * 2 = 2
2x
Inner
3 * 1 = 3
3x
Last
3 * 2 = 6
6
Combine Like Terms
2x + 3x = 5x
x^2 + 5x + 6
(a + b)^2 = a^2 + 2ab + b^2
(a - b)^2 = a^2 - 2ab + b^2
(a + b)(a - b) = a^2 - b^2
(x + a)(x + b) = x^2 + (a+b)x + ab
Product
x^2 + 5x + 6
(x + 3) * (x + 2)
Quick-start with common scenarios
Test your skills with practice problems
Practice with 3 problems to test your understanding.
FOIL is a method to multiply two binomials: (a + b)(c + d). FOIL stands for First, Outer, Inner, Last. Multiply: First terms (a times c), Outer terms (a times d), Inner terms (b times c), Last terms (b times d). Then add: ac + ad + bc + bd. Example: (x + 3)(x + 2) = x^2 + 2x + 3x + 6 = x^2 + 5x + 6.
FOIL is a mnemonic for multiplying two binomials (expressions with two terms each). The letters stand for First, Outer, Inner, Last, representing which terms to multiply together. When multiplying (a + b)(c + d), you multiply the First terms (a and c), the Outer terms (a and d), the Inner terms (b and c), and the Last terms (b and d), then add all products together. FOIL is actually just a special case of the distributive property.
FOIL is an acronym for First, Outer, Inner, Last. It describes the order in which you multiply terms when expanding the product of two binomials: (a + b)(c + d). First means multiply a times c, Outer means a times d, Inner means b times c, and Last means b times d.
Step 1: Multiply the First terms (first term of each binomial). Step 2: Multiply the Outer terms (first of first, last of second). Step 3: Multiply the Inner terms (last of first, first of second). Step 4: Multiply the Last terms (last term of each). Step 5: Add all four products. Step 6: Combine like terms to simplify.
FOIL specifically works for multiplying two binomials (expressions with exactly two terms each). For multiplying a binomial by a trinomial, or two trinomials, you need to use the full distributive property, multiplying each term in the first polynomial by each term in the second. FOIL is really just a mnemonic for the distributive property applied to binomials.
Three important patterns: (1) Perfect square: (a + b)^2 = a^2 + 2ab + b^2, (2) Difference of squares: (a + b)(a - b) = a^2 - b^2, (3) Standard trinomial: (x + a)(x + b) = x^2 + (a+b)x + ab. Recognizing these patterns speeds up multiplication and factoring.
The Outer and Inner products often contain like terms (same variable to the same power). For example, in (x + 3)(x + 2), the Outer is 2x and Inner is 3x. Since both are "x to the first power" terms, they combine to give 5x. This is why the result is usually a trinomial rather than four terms.
FOIL is really just a specific application of the distributive property. When you distribute (a + b)(c + d), you get a(c + d) + b(c + d) = ac + ad + bc + bd. This is exactly what FOIL produces. FOIL is simply a mnemonic to help remember all four multiplications.
The reverse of FOIL is factoring. If you have a trinomial like x^2 + 5x + 6, you can factor it back into (x + 2)(x + 3). To factor x^2 + bx + c, find two numbers that multiply to c and add to b. Those become the constants in your binomial factors.
Yes, FOIL works with any numbers including negatives. Just be careful with signs. For example, (x - 3)(x + 2): First = x^2, Outer = 2x, Inner = -3x, Last = -6. Result: x^2 + 2x - 3x - 6 = x^2 - x - 6. Remember that minus times plus gives minus.
Last updated: 2025-01-15
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Product
x^2 + 5x + 6
(x + 3) * (x + 2)