Algebra - GED Math

Let be the number of dimes. Then there are nickels.

An equation can be set up and solved for for the amount of money:

, the number of dimes.

Note the switch in the inequality symbol.

.

This can also be written as .

Note that the inequality symbol changes.

or, in interval notation, .

Multiply both sides by 4 to isolate :

is undefined if and only if its denominator is equal to 0.

Take the product of the two binomials using the FOIL method.

First:

Outer:

Inner:

Last:

Add these:

Collect the like terms in the middle by subtracting coefficients:

Therefore,

Distribute the by multiplying it by each term:

,

the correct product.

Expand the following expression:

This is what is known as a difference of squares. We can use FOIL to find our answer.

Remember FOIL? First Outer Inner Last

This means we need to multiply each pair of terms in parentheses to get the correct answer:

First:

Outer:

Inner:

Last:

Put it all together to get:

Making our answer

Do you see why it's called a difference of squares?

Foiling means to take two equations and merge them into one. It's also the same as saying you want to multiply one equation with another, which is what we'll be doing.

Our two equations are and , which is the same as . In order to multiply these two equations together, you must first multiply the first unit in your equation with everything in the second equation, then the second unit with everything in your second equation.

Multiply the in the first equation with the from the second equation:

Multiply the in the first equation with the from the second equation:

Now multiply the with the from the second equation:

Multiply the from the first equation with the from the second equation:

We won't multiply the second equation with the first one like we did above, as that would give us the same answers. Take all the answers you got from above and now string them together like so:

We're almost done, but we seem to be have more than one ; and . These two terms can be combined like so:

Since nothing else seems to have more than one of itself, we can now put the equation together. Make sure to go in order of highest power of to the lowest power of .

Your answer is

Note that the inequality symbol changes.

or, in interval notation, .

In order to solve for , we will need the equation to be in terms of , and isolate the variable .

Solve by grouping the terms together. Subtract on both sides.

Divide by negative five on both sides.

The answer is:

Distribute the term on the right side of the equation.

Combine like terms.

Subtract on both sides.

Divide by negative on both sides.

The answer is:

Every second, you know that Marge and William will become a total of or . Now, you can use the simple work formula for distance:

or, for our data,

(Remember kilometers is meters.)

Thus, solving for you get:

This is in seconds, though. You need minutes. To convert, you need to divide by :

Timmy makes ten dollars per hour for eight hours.

For five days:

Timmy also will pay for the week to get to work and back.

Subtract his expense from his earnings for the week.

Timmy will have by the end of the week.

where

The numbers and fit those criteria. Therefore,

You can double check the answer using the FOIL method

Perhaps the easiest way to identify the factor is to take advantage of the factor theorem, which states that is a factor of polynomial if and only if . We substitute 1, 2, 4, and 9 for in the polynomial to identify the factor.

:

:

:

:

Only makes the polynomial equal to 0, so among the choices, only is a factor.

Perhaps the easiest way to identify the factor is to take advantage of the factor theorem, which states that is a factor of polynomial if and only if . We substitute and for in the polynomial to identify the factor.

:

:

:

:

Only makes the polynomial equal to 0, so of the four choices, only is a factor of the polynomial.

Factor all the way to its prime factorization.

can be factored as the difference of two perfect square terms as follows:

is a factor, and, as the sum of squares, it is a prime. is also a factor, but it is not a prime factor - it can be factored as the difference of two perfect square terms. We continue:

Therefore, all of the given polynomials are factors of , but is the correct choice, as it is not a prime factor.