FOIL Method

FOIL is a mnemonic for the standard method of multiplying two binomials. It's a handy way to remember how to multiply two binomials. This method is referred to as a FOIL method and is commonly used in linear binomials. FOIL is the acronym of "first, outside, inside, last."

• First: multiply the first terms in each set of parentheses.
• Outside: multiply the two terms on the outside.
• Inside/Inner: multiply both of the inside terms.
• Last: multiply the last terms in each set of parentheses.

For \(( 2x -5) (x -4),\) we have

• first: \(2x \times x=2x^2\)
• outside: \(2x \times (−4)= −8x\)
• inside: \(−5 \times x= −5x\)
• last: \((−5) \times (−4)= 20.\)

For \(( x + 3 ) ( x + 5 ),\) FOIL gives

\[\begin{align} ( x + 3 ) ( x + 5 ) &= x \cdot x + x \cdot 5 + 3 \cdot x + 3 \cdot 5\\ &= x^2 + 5x + 3x + 15\\ &=x^2 + 8x + 15. \end{align}\]

For \((2x + 3) (x - 5),\) FOIL gives the following:

• First: multiply \(2x\) with \(x\) to get \(2x^2.\)
• Outside: multiply \(2x\) with \(-5\) to get \(-10x.\)
• Inside: multiply \(3\) with \(x\) to get \(3x.\)
• Last: multiply \(3\) with \(-5\) to get \(-15.\)

Therefore, \((2x + 3) (x - 5) = 2x^2-7x-15.\)

The FOIL rule converts a product of two binomials into a sum of four monomials/power products. This reverse process is called factorization.