This is the difference of squares.  You must know this formula for the GRE!

a² – b² = (a – b)(a + b)

Here a = 5x and b = 6y, so the difference of squares formula gives us (5x – 6y)(5x + 6y).

First pull out 3u from both terms.

3u⁴ – 24uv³ = 3u(u³ – 8v³) = 3u[u³ – (2v)³]

This is a difference of cubes. You will see this type of factoring if you get to the challenging questions on the GRE. They can be a pain to remember but pat yourself on the back for getting to such hard questions! The difference of cubes formula to remember is a³ – b³ = (a – b)(a² + ab + b²). In our problem, a = u and b = 2v, so 

3u⁴ – 24uv³ = 3u(u³ – 8v³) = 3u[u³ – (2v)³]

                = 3u(u – 2v)(u² + 2uv + 4v²)

To begin, let's factor the first two terms and the second two terms separately.  

z³ – z² – 9z + 9 = (z³ – z²) + (–9z + 9) = z²(z – 1) – 9(z – 1)

(z – 1) can be pulled out because it appears in both terms. 

z³ – z² – 9z + 9 = (z³ – z²) + (–9z + 9) = z²(z – 1) –  9(z – 1) = (z – 1)(z² – 9)

(z² – 9) is a difference of squares, so we can use the formula a² – b² = (a – b)(a + b).

z³ – z² – 9z + 9 = (z³ – z²) + (–9z + 9) = z²(z – 1) – 9(z – 1)

                     = (z – 1)(z² – 9)

                     = (z – 1)(z – 3)(z + 3)

We know the equation a² – b² = (a + b)(a – b) for the difference of squares. Since y² is the square of y, and 4 is the square of 2, the correct answer is  (y + 2)(y – 2).

Factor the equation by finding two numbers that add to -3 and multiply to -28.

Factors of 28: 1,2,4,7,14,28

-7 and 4 work.

(x-7)(x+4) = 0

Set each equal to zero:

x=7,-4

Since  is positive, we can divide both sides of the inequality by :

 or .

For the second equation, solve for  in terms of .

Plug this value of y into the first equation.

Don't forget to switch the inequality direction if you multiply or divide by a negative.

Now that we have the equation in slope-intercept form, we can see that the y-intercept is 6.

Begin by adding  to both sides, this will get the variable isolated:

Or...

Next, divide both sides by :

Notice that when you divide by a negative number, you need to flip the inequality sign!

Choice a is equivalent because we can say that technically we are multiplying two fractions together: (xy)/z and (5(x + y))/1.  We multiply the numerators together and the denominators together and end up with xy (5x + 5y)/z.  xy (5y + 5x)/z is also equivalent because it is only simplifying what is inside the parentheses and switching the order- the commutative property tells us this is still the same expression.  5x²y + 5xy²/z is equivalent as it is just a simplified version when the numerators are multiplied out.  Choice 5x² + y²/z is not equivalent because it does not account for all the variables that were in the given expression and it does not use FOIL correctly.