Question

In your own words, describe the process of multiplying binomials using the FOIL method. You can either write a paragraph describing the process, or you can list a set of steps to follow. Include an example problem in your answer.

Answer 1

Answer:

We can describe FOIL using the distributive property of multiplication. For real numbers x, y, and z the distributive property of multiplication states that

$(x + y)z = xz + yz$

Now let $z = v + w$, where v and w are real numbers.

Then $(x + y)(v + w) = x(v + w) + y(v + w)$

Here, we can again use the distributive property of multiplication to get

$x(v + w) + y(v + w) = xv + xw + yv + yw$

Hence, it follows that

$(x + y)(v + w) = xv + xw + yv + yw$

This is the method of FOIL (first: xv, outside: xw, inside: yv, last: yw)

For example, $(1 + 2)(3 + 4) = 21 \\\text{and} \\1(3) + 1(4) + 2(3) + 2(4) = 3 + 4 + 6 + 8 = 21$