To factor the polynomial, we need the numbers that multiply to –12 and add to +1. This leads us to –3 and +4. We solve the polynomial by setting it equal to 0.

So either x = 3 or x = –4 will make the product equal to 0.

This is a factoring problem, so we need to get all of the variables on one side and set the equation equal to zero. To do this we subtract 128 from both sides to get .

We then notice that all four numbers are divisible by four, so we can simplify the expression to .

Think of the equation in this format to help with the following explanation.

We must then factor to find the solutions for x. To do this we must make a factor tree of c (which is 32 in this case) to find the possible solutions. The possible numbers are 1 * 32, 2 * 16, and 4 * 8.

Since c is negative, we know that our factoring will produce a positive and negative number.

We then look at b to see if the greater number will be positive or negative. Since b is positive, we know that the greater number from our factoring tree will be positive.

We then use addition and subtraction with the factoring tree to find the numbers that add together to equal b. Remember that the greater number is positive and the lesser number is negative in this example.

Positive 8 and negative 4 equal b. We then plug our numbers into the factored form of .

We know that anything multiplied by 0 is equal to 0, so we plug in the numbers for x which make each equation equal to 0. In this case .

The quadratic can be solved as . Setting each factor to zero yields the answers.

To factor this, we need to find a pair of numbers that multiplies to 6 and sums to 5. The numbers 2 and 3 work. (2 * 3 = 6 and 2 + 3 = 5)

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

x = –2 or x = –3

To factor a trinomial without using the quadratic equation, a few basic steps can be taken. The first step is always to rearrange our trinomial into quadratic form.

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First, create two blank binomials.

Start by factoring our first term back into the first term of each biniomial. Since the only reasonable roots of are and , we know that

Next, factor out our constant, ignoring the sign for now. The factors of are either and , and , or and , We must select those factors which have either a difference or a sum equal to the value of in our trinomial. In this case, neither and nor and can sum to , but and can. Now, we can add in our missing values:

One last step remains. We must check our signs. Since is positive in our trinomial, Either both signs are negative or both are positive. To figure out which, check the sign of in our trinomial. Since is negative, both signs in our binomials must be negative.

Thus, our two binomial factors are and . Note that this can also be written as , and if you graph this, you get a result identical to .

To find the domain, find all areas of the number line where the fraction is defined.

because the denominator of a fraction must be nonzero.

Factor by finding two numbers that sum to -2 and multiply to 1. These numbers are -1 and -1.

Given the equation

Realizing that the domain is restricted by the denominator, meaning that the denominator can not be equal to 0.

Set the denominator = to 0 and solve.

First factoring ,

Zero-product Property, setting both quantities equal to 0 and solve.

So when x is 6 or -1, our denominator will be 0. Meaning those would be our domain restrictions.

Given the following expression:

We need to find factors of that add up to .

can be broken down into the following factors:

Of these choices, only adds up to . Additionally, the coefficient in front of the variable is , so we do not need to worry about that when finding these values. There are no negatives in the quadratic expression, so the signs in the factored form are all positive. This gives us the final answer of

You can use the FOIL method to re-expand the expression and check your work!

The expression can be factored by finding terms that multiply back to the original expression. The easiest way is to find two numbers that add to the middle term as well as multiply to the last term The numbers that satisfy both of these conditions are and , so the answer is .

To factor a trinomial without using the quadratic equation, a few basic steps can be taken. The first step is always to rearrange our trinomial into quadratic form, but this is already done.

First, create two blank binomials.

Start by factoring our first term back into the first term of each biniomial. Since the only reasonable roots of are and , we know that

Next, factor out our constant, ignoring the sign for now. The factors of are either and , and , or and , We must select those factors which have either a difference or a sum equal to the value of in our trinomial. In this case, neither and nor and can sum to , but and can. Now, we can add in our missing values:

One last step remains. We must check our signs. Since is positive in our trinomial, Either both signs are negative or both are positive. To figure out which, check the sign of in our trinomial. Since is negative, both signs in our binomials must be negative.

Thus, our two binomial factors are and .