Make sure you keep track of negative signs when doing FOIL, especially when doing the Outer and Inner steps.

When the 2 terms differ only in their sign, the \(\displaystyle x\)-term drops out from the final product.

FOIL is a helpful way to remember how to multiply terms when we are multiplying two binomials.

F stands for first. We multiply the two "first" terms in each of the binomials.

O stands for outer. We multiply the outermost terms in the expression.

I stands for inner. We multiply the innermost terms in the expression.

L stands for last. We multiply the two "last" terms in each of the binomials.

F - \(\displaystyle \small \small 5x *3x=15x^{2}\)

O - \(\displaystyle \small \small 5x*y=5xy\)

I - \(\displaystyle \small \small -2y*3x=-6xy\)

L - \(\displaystyle \small \small -2y*y=-2y^{2}\)

Now we combine each part using addition to get:

\(\displaystyle \small 15x^{2}-6xy+5xy-2y^{2}\)

The two terms in the middle can be combined to get our final answer:

\(\displaystyle \small 15x^{2}-xy-2y^{2}\)

Recall that using FOIL means multiplying the first terms, then the outside terms, next the inside terms, and finally, the last terms. Therefore, after multiplying all of those, you should get: \(\displaystyle -12x^2+8x+3x-2\). Combine your like terms to get a final answer of: \(\displaystyle -12x^2+11x-2\).

Recall that FOIL means multiplying the first terms, then outside terms, next inside terms, and then the last terms. Therefore, when you do those steps, you get: \(\displaystyle 12x^2+16x-3x-4\). Simplify the middle terms to get your answer of \(\displaystyle 12x^2+13x-4\).

To solve this problem, recall that you will be using the FOIL method to mutlply these binomials: \(\displaystyle (4-2x)(4-2x)=16-8x-8x+4x^2\). Then, combine like terms to get your answer: \(\displaystyle 4x^2-16x+16\).

To multiply these binomials, use the FOIL method: \(\displaystyle 18x^2+24x-3x-4\). Combine like terms to get your answer of \(\displaystyle 18x^2+21x-4\).

\(\displaystyle Profit = (Cookies\: Sold) \times (Price\: per\: Cookie)\)

This function can be expanded as

\(\displaystyle Profit=(150+5x)(1-.05x)\)

where

x=number of $0.05 changes to the initial price which will maximize potential revenue.

Since we are seeking to maximize  profit, we need to find the point in this function, where a given number, x, of changes to the original price yields the greatest additional profit by balancing the additional revenue per cookie with the $0.05 decrease in cost.

In other words, we are seeking the vertex of this parabola.

First we convert the profit function into quadratic form by FOILing:

\(\displaystyle Profit=150-7.5x+5x-.25x^2\)

rearranging this, we find that:

\(\displaystyle Profit=-.25x^2-2.5x+150\)

Recall that

\(\displaystyle h=\frac{-b}{2a} =\frac{2.5}{2(-.25)}=\frac{2.5}{.5}=5\)

So, in order to maximize profits, we should reduce each cookie's cost by $.05 5 times.

\(\displaystyle Cost \: Per \: Cookie = 1-(.05\times5) = 1-.25 = 0.75\)

We should sell each cookie for $0.75

To FOIL (First, Outer, Inner, Last), we start be multiplying the first terms together:

\(\displaystyle 3x^{2}\cdot 2x = 6x^{3}\)

Then the Outer terms:

\(\displaystyle 3x^{2} \cdot -5 = -15x^{2}\)

Then the Inner terms:

\(\displaystyle 2 \cdot 2x = 4x\)

And finally the Last terms:

\(\displaystyle 2 \cdot -5 = -10\)

We then collect each of the answers and put them in descending order of degree of their exponent:

\(\displaystyle f(x)=6x^{3}-15x^{2}+4x-10\)

To FOIL (First, Outer, Inner, Last), we start by multiplying the first term in each grouping together:

\(\displaystyle x \times x = x^2\)

Then we multiply the outer terms together (the first term in the first grouping, the last term in the second grouping):

\(\displaystyle x \times -7 = -7x\)

Then we multiply the inner terms together (the second term in the first grouping, the first term in the second grouping):

\(\displaystyle 4 \times x =4x\)

And finally the last term in each grouping:

\(\displaystyle 4 \times -7 = -28\)

We can then collect everything back into our function:

\(\displaystyle f(x)=x^2 - 7x + 4x - 28\)

And combine like terms (in this case, both terms with an \(\displaystyle x\) in them):

\(\displaystyle f(x)= x^2-3x-28\)