1. ## ACT Math Section

I wasn't sure whether this belonged in this section or the algebra section. Anyway, I'm studying for the math portion of the ACT and I can't remember how to solve certain problems. Like this one:

Which of the following is NOT a factor of z^5 - 16z ?

A. z^2 - 1
B. z^2 - 4
C. z + 2
D. z
E. z - 2

Now I'm aware the answer is A but I have no idea why. If someone could provide an explanation for how to solve that would be great.

Another One: If (2x - y) / (x + y) = 2/3, then x/y = ?

(2x - y is the numerator, x + y is the denominator)

F. 1/2
G. 2/3
H. 5/4
J. 5/3
K. 5

One more: For all nonzero x, y, and z such that x = yz, which of the following must be equivalent to xy?

A. z/x
B. yz^2
C. yz
D. x^2/z
E. x/y

I'm sorry if I posted in the wrong place. Thanks in advance.

2. You should post your attempts to each question.

$z^5-16z$

Take out a common factor of z.

$z(z^4-16)$

$z((z^2)^2-4^2)$

By difference of 2 squares

$z(z^2-4)(z^2+4)$

$z(z-2)(z+2)(z^2+4)$

$z(z-2)(z+2)(z^2-(\sqrt{-4})^2)$

$z(z-2)(z+2)(z^2-(\sqrt{4}i)^2)$

$z(z-2)(z+2)(z-\sqrt{4}i)(z+\sqrt{4}i)$

$z(z-2)(z+2)(z-2i)(z+2i)$

3. Originally Posted by 5441lk
I wasn't sure whether this belonged in this section or the algebra section. Anyway, I'm studying for the math portion of the ACT and I can't remember how to solve certain problems. Like this one:

Which of the following is NOT a factor of z^5 - 16z ?

A. z^2 - 1
B. z^2 - 4
C. z + 2
D. z
E. z - 2

Now I'm aware the answer is A but I have no idea why. If someone could provide an explanation for how to solve that would be great.

Another One: If (2x - y) / (x + y) = 2/3, then x/y = ?

(2x - y is the numerator, x + y is the denominator)

F. 1/2
G. 2/3
H. 5/4
J. 5/3
K. 5
Multiplying on both sides by 3(x+y), 3(2x-y)= 2(x+y) so 6x- 3y= 2x+ 2y. Adding 3y and subtracting 2x from both sides, 4x= 5y. Now, what is x/y?

One more: For all nonzero x, y, and z such that x = yz, which of the following must be equivalent to xy?

A. z/x
B. yz^2
C. yz
D. x^2/z
E. x/y

I'm sorry if I posted in the wrong place. Thanks in advance.
The obvious first step is to multiply on both sides by y to get $xy= y^2z$. But that isn't any of the given answers so lets try eliminating some: if y= 2 and z= 1, then x= (2)(1)= 2 and xy= 4. z/x= 2/1= 2, not 4 so it can't be that. $yz^2= 2(1^2)= 2$ so that is not it. yz= 2(1)= 2 so that is not it. $x^2/z= 2^2/1= 4$ so that might be it! x/y= 2/2= 1 so that is not it.

If this were a multiple choice test where I did not have much time, I think I could, with confidence, mark "D" and go on. But, of course, just showing it is correct for one chosen value of y and z doesn't mean it is always true. How can we get from x=yz to $xy= x^2/z$? As before, multiplying both sides of x= yz by y gives $xy= y^2z$. Now that I see the "correct" answer has only x and z in it, I can argue that x= yz is equivalent to y= x/z (divide both sides by z) and so $y^2= x^2/z^2$. Then $xy= y^2z= (x^2/z^2)a= x^2/z$ as claimed.

4. A more simple approach to the last one is to simply note that $x=yz \Rightarrow y = \frac{x}{z}$ now substitute this with $y$ in the expression $xy$ to get: $xy = x\frac{x}{z} = \frac{x^2}{z}$.

5. [quote=pickslides;357360]You should post your attempts to each question.

Ok, next time I will.