💡 What You Will Learn
A binomial is a polynomial with two terms, like (5x − 1) or (x + 3). Multiplying two binomials requires multiplying every term in the first by every term in the second. The FOIL method — First, Outer, Inner, Last — gives you a systematic order so you never miss a term!
Binomial: A polynomial with exactly two terms separated by + or −.
Examples: (5x − 1), (x + 3), (2x + 7), (x − 4)
FOIL: An acronym for the order in which you multiply the terms of two binomials:
First, Outer, Inner, Last
Each binomial has 2 terms. 2 × 2 = 4 multiplications needed. FOIL gives you all 4 in order — none missed!
After FOIL you get up to 4 terms. Combine the "like terms" (same variable and exponent) to simplify to the final answer.
First Outer Inner Last
x · x = x² (first term of left × first term of right)
x · 5 = 5x (first term of left × last term of right)
3 · x = 3x (last term of left × first term of right)
3 · 5 = 15 (last term of left × last term of right)
= x² + 8x + 15
5x and 3x are "like terms" — same variable (x) and same exponent (1). Combine them by adding their coefficients: 5 + 3 = 8. So 5x + 3x = 8x.
25x² + (−5x) + (−5x) + 1
−5x + (−5x) = −10x
| Choice | What Went Wrong | Correct? |
|---|---|---|
| 25x² − 10x + 1 | Full FOIL correctly applied ✅ | ✅ CORRECT |
| 25x² + 1 | Forgot the middle terms (Outer + Inner) ❌ | ❌ Trap! |
| 25x² − 1 | Forgot middle terms; (−1)(−1) gave −1 instead of +1 ❌ | ❌ Trap! |
| 25x² − 2x + 1 | Used coefficient 1 for x terms instead of 5 ❌ | ❌ Wrong |
The most common mistake is writing just 25x² + 1 by squaring each term separately: (5x)² = 25x², (−1)² = 1. This IGNORES the outer and inner products! You must use FOIL to get all 4 terms.
Perfect Square: When both binomials are identical — the middle term doubles!
Difference of Squares: When one has + and other has − same terms — middle terms cancel, leaving just a² − b²!
Recognizing these patterns saves time on the GED.
The #1 trap is just squaring each term: (5x)² + (−1)² = 25x² + 1. WRONG! You must do F, O, I, AND L to get all four products.
(−1)(−1) = +1. This is why the Last term in (5x−1)(5x−1) is +1, not −1. Watch your signs on the L step!
x² and x are NOT like terms. Only combine terms with the exact same variable and exponent. 25x² stays separate from −10x.
Substitute x = 1 into both the original and your answer. (5(1)−1)(5(1)−1) = (4)(4) = 16. Answer: 25(1)²−10(1)+1 = 16 ✅
| Letter | Stands For | Which Terms | In (5x−1)(5x−1) |
|---|---|---|---|
| F | First | 1st × 1st | 5x · 5x = 25x² |
| O | Outer | 1st × 2nd | 5x · (−1) = −5x |
| I | Inner | 2nd × 1st | (−1) · 5x = −5x |
| L | Last | 2nd × 2nd | (−1) · (−1) = +1 |