Review and Other Topics - Math

This is a difference of perfect cubes so it factors to . Only the first expression will yield an answer when set equal to 0, which is 1. The second expression will never cross the -axis. Therefore, your answer is only 1.

Factor the equation to . Set and get one of your 's to be . Then factor the second expression to . Set them equal to zero and you get .

Use the difference of perfect cubes equation:

In ,

and

To factor and solve for in the equation

Factor out of the equation

Use the "difference of squares" technique to factor the parenthetical term, which provides the completely factored equation:

Any value that causes any one of the three terms , , and to be will be a solution to the equation, therefore

You can see that each term in the equation has an "x", therefore by factoring "x" from each term you can get that the equation equals .

First consider all the factors of 12:

1 and 12

2 and 6

3 and 4

Then consider which of these pairs adds up to 7. This pair is 3 and 4.

Therefore the answer is .

Begin by extracting from the polynomial:

Now, factor the remainder of the polynomial as a difference of cubes:

Begin by rearranging like terms:

Now, factor out like terms:

Rearrange the polynomial:

Begin by rearranging like terms:

Now, factor out like terms:

Rearrange the polynomial:

Factor:

Begin by separating into like terms. You do this by multiplying and , then finding factors which sum to

Now, extract like terms:

Simplify:

To begin, distribute the squares:

Now, combine like terms:

Begin by extracting from the polynomial:

Now, distribute the cubic polynomial:

Begin by extracting like terms:

Now, rearrange the right side of the polynomial by reversing the signs:

Combine like terms:

Factor the square and cubic polynomial:

Begin by rearranging the terms to group together the quadratic:

Now, convert the quadratic into a square:

Finally, distribute the :

Begin by extracting from the polynomial:

Now, rearrange to combine like terms:

Extract the like terms and factor the cubic:

Simplify by combining like terms:

Begin by extracting from the polynomial:

Now, rearrange to combine like terms:

Extract the like terms and factor the cubic:

Simplify by combining like terms:

You can see that each term in the equation has an "x", therefore by factoring "x" from each term you can get that the equation equals .

To factor and solve for in the equation

Factor out of the equation

Use the "difference of squares" technique to factor the parenthetical term, which provides the completely factored equation:

Any value that causes any one of the three terms , , and to be will be a solution to the equation, therefore

This is a difference of perfect cubes so it factors to . Only the first expression will yield an answer when set equal to 0, which is 1. The second expression will never cross the -axis. Therefore, your answer is only 1.

Factor the equation to . Set and get one of your 's to be . Then factor the second expression to . Set them equal to zero and you get .