Algrebra1, I don't get factoring or foiling and unfoiling at all?

Question

My teacher has gave us an assignment with factoring, foiling and unfoiling. He doesn't know how to teach, so I have no clue what's going on. Here's some examples of some questions:

1.)     x{squared} -12x+0=0

2.)    10x{squared} +13x+4=0

3.)      -8x{squared} +15x-7=0

 

He has us do 18 of these, can I have some help?

Answer 1

Hi, there!

Let's start with the first example. When you have a variable in each term of your equation, you can factor the variable out. For example:

17x² + 14x + 0 = 0

This can be simplified to 17x² + 14x = 0.

Then, it becomes clear that both of your terms include an x in them, so you can take that x out like this:

x(17x + 14) = 0.

Now the equation is factored.

Note that if you multiplied the outside x by each of the terms in the parentheses, you'd get your original equation back: x(17x) = 17x². x(14) = 14x. When you add these together, you get 17x² + 14x = 0

Now let's move to the next type of example you gave.

When one of the terms in your equation is just a number with no variable, then you have to reverse FOIL. FOIL stands for First Outer Inner Last.

Let's start with a FOIL problem.

(3x + 5)(x + 2) = 0

To foil this equation, we start by multiplying the First term of each set of parentheses:

(3x + 5)(x + 2) = 0

3x(x) = 3x².

Next, we multiply the Outer terms of the parentheses:

(3x + 5)(x + 2) = 0

3x(2) = 6x

Then, we multiply the Inner terms of the parentheses:

(3x + 5)(x + 2) = 0

5(x) = 5x

Finally, we multiply the Last terms of each set of parentheses:

(3x + 5)(x + 2) = 0

5(2) = 10

Now we add up all these terms: 3x² + 6x + 5x + 10 = 0.

Since 6x and 5x can be added together, the completely foiled equation is 3x² + 11x + 10 = 0.

Let's get back to the kind of problem you are dealing with in your homework.

If we start with the equation 3x² + 11x + 10 = 0, now we have to unfoil it or factor it - figure out which terms go in the parentheses to give us this equation.

We know that we'll start with the First term of each set of parentheses to get 3x². The only way we can get x² is if both terms have an x in them since x(x) = x²:

(_x +/- _)(_x +/- _) = 0

Now we have to figure out how to find two numbers that multiply to 3. Since 3 is prime, the only possibilities are 1 and 3: 1(3) = 3. So we can fill in these numbers for our First terms.

(1x +/- _)(3x +/- _) = 0

Since 1x = x, it's neater to just write x by itself, so we have

(x +/- _)(3x +/- _) = 0

We could also write this as (3x +/- _)(x +/- _). Both are correct because x(3x) = (3x)x = 3x².

The next step I take is I look at the last term of 3x² + 11x + 10 = 0. The last term is 10. That means that our next numbers have to multiply together to equal 10, so our choices are either 1 & 10 or 2 & 5.

(_ +/- 1)(_ +/- 10) or (_ +/- 2)(_ +/- 5).

Since our last term is +10 and not -10, we need our last numbers in the parentheses to both be positive or both be negative so that they multiply to positive 10.

(_ + _)(_ + _) or (_ - _)(_ - _).

Looking at our equation again, 3x² + 11x + 10 = 0, the middle term is +11x. This means that our inner and outer steps must add to a positive number. Our choices from the last step mean we can either add two positive numbers or two negative numbers. Adding two positive numbers gives us a positive answer. Adding two negative answers gives us a negative answer. Therefore, we need to use two positive numbers.

(_ + _)(_ + _)

Now we have to figure out which pairs of numbers will give us 11x in the middle. Eventually this process will become easier as you get more practice with it, so it will take less time, but for now, let's look at all our possibilities:

a. (x + 1)(3x +10)

b. (x + 10)(3x + 1)

c. (x + 2)(3x + 5)

d. (x + 5)(3x + 2)

Let's add the Outer and Inner results from each one and see which gives us +11.

a. x(10) + 1(3x) = 10x + 3x = 13x

b. x(1) + 10(3x) = x + 30x = 31x

c. x(5) + 2(3x) = 5x + 6x = 11x

d. x(2) + 5(3x) = 2x + 15x = 17x

The factorization of 3x² + 11x + 10 = 0 is therefore (x + 2)(3x + 5) = 0.

Note that the order does not matter. If you put (3x + 5)(x + 2) = 0, this is still correct. The important thing is that your factors are (x + 2) together and (3x + 5) together.

Let's look at your last example now. This time, the highest x value is negative. This makes it difficult to factor if you don't deal with the negative sign first. Let's say you have the equation -4x² + 12x - 8 = 0.

There are two ways you can fix this equation. You can either (a) multiply the entire equation by (-1) or (b) bring all the numbers to the other side of the equation.

(a) (-1) [-4x² + 12x - 8 = 0]

(-1)(-4x²) = 4x²

(-1)(12x) = -12x

(-1)(-8) = 8

(-1)(0) = 0

4x² - 12x + 8 = 0

(b) -4x² + 12x - 8 = 0

12x - 8 = 0 + 4x²

-8 = 4x² - 12x

0 = 4x² - 12x + 8

Either way, all we're doing is switching each positive and negative sign so that the term with the highest degree of x, in this case 4x², is a positive number. Then we factor the same way we did in the previous example. In the end, this gives us (x - 1)(4x - 8).

Each of these example equations had a positive last term (either +10 or +8). Because of this, the last term in both sets of parentheses was either positive (+2, +5) or negative (-1, -8). Because both were positive or both were negative, the two terms could be added to give you the middle term of the equation (either +11x or -12x in these examples).

If the last term of the equation is negative, then the last term of one set of parentheses must be positive and the other must be negative. This is because the only way to multiply two numbers and get a negative answer is to multiply a positive number by a negative number.

This will give you (__ + __)(__ - __) or (__ - __)(___ + ___).

This means that when you multiply Outer terms and then Inner terms, one answer will be positive, and one will be negative, so instead of adding, you will be subtracting. You'll have to arrange your numbers in the parentheses so that your answer is positive if the middle term of your equation is positive, or negative if the middle term is negative. For example:

(2x + 3)(6x - 5)

Outer: 2x(-5) = -10x

Inner: 3(6x) = +18x

-10x + 18x = +8x

This matches a positive middle term in the equation 12x² + 8x - 15 = 0.

If we switch the positive and negative signs, we get this:

(2x - 3)(6x + 5)

Outer: 2x(5) = +10x

Inner: (-3)(6x) = -18x

10x - 18x = -8x

This matches a negative middle term in the equation 12x² - 8x -15 = 0.

One last note: always make sure before trying to factor an equation that you have an expression equal to 0.

If you have something like 12x² - 8x = 15, bring one side over so that you're back to an equation equal to 0. Make this equation 12x² - 8x - 15 = 0, and then start factoring it.

To summarize:

1) Make sure you have an expression equal to 0.

2) Make sure your highest x term is positive.

3) Make sure you don't have any extra x's (see first example).

4) Start unFOIL-ing the equation, paying attention to the positive and negative signs of the middle and last terms of the equation.

5) You can always check your work by FOIL-ing your answer to see if you get the equation you started with (see second example).

Good luck!