**Problem 1**

What is the product of 2*x *+ 3 and 4*x*^{2} – 5*x +* 6?

(1) 8*x*^{3} – 2*x*^{2} + 3*x *+ 18

(2) 8*x*^{3} – 2*x*^{2} – 3*x *+ 18

(3) 8*x*^{3} + 2*x*^{2} – 3*x *+ 18

(4) 8*x*^{3} + 2*x*^{2} + 3*x *+ 18

¿Cuál es el producto de 2*x *+ 3 y 4*x*^{2} – 5*x *+ 6?

(1) 8*x*^{3} – 2*x*^{2} + 3*x *+ 18

(2) 8*x*^{3} – 2*x*^{2} – 3*x *+ 18

(3) 8*x*^{3} + 2*x*^{2} – 3*x *+ 18

(4) 8*x*^{3} + 2*x*^{2} + 3*x *+ 18

*Discussion and Solution:*

To solve this problem, first multiply the second expression by 2*x*. Then multiply the second expression by the 3. Finally, combine the like terms.

2*x*(4*x*^{2} – 5*x* + 6) = 8*x*^{3} – 10*x*^{2} + 10*x*

3(4*x*^{2} – 5*x* + 6) = 12*x*^{2} – 15*x* + 18

(8*x*^{3} – 10*x*^{2} + 10*x*) + (12*x*^{2} – 15*x* + 18)

8*x*^{3} + 2*x*^{2} – 3*x* + 18

Choice (3).

**Problem 2**

What are the solutions to the equation 3*x*^{2} + 10*x *= 8?

(1) 2/3 and – 4

(2) – 2/3 and 4

(3) 4/3 and – 2

(4) – 4/3 and 2

¿Cuáles son las soluciones para la ecuación 3*x*^{2} + 10*x *= 8?

(1) 2/3 and – 4

(2) – 2/3 and 4

(3) 4/3 and – 2

(4) – 4/3 and 2

*Discussion and Solution:*

To solve this equation, first subtract 8 from each side.

3*x*^{2} + 10*x* – 8 = 8 – 8

3*x*^{2} + 10*x* – 8 = 0

Since the only factors for the coefficient 3 are 3 and 1, the first components for the factors are:

(3*x* )(*x* )

Since there is a negative sign in front of the 8, we get:

(3*x* + )(*x* – ) or (3*x* – )(*x* + )

The only factors for the 8 are (8 and 1) and (4 and 2).

That gives us the following as possible solutions:

(3*x* + 8)(*x* – 1)

(3*x* – 8)(*x* + 1)

(3*x* + 4)(*x* – 2)

(3*x* + 2)(*x* – 4)

(3*x* – 4)(*x* + 2)

(3*x* – 2)(*x* + 4)

Using FOIL, the only one of the above expressions that would result in

3*x*^{2} + 10*x* – 8 is:

(3*x* – 2)(*x* + 4)

Setting each equal to 0 and solving, we get:

(3*x* – 2) = 0

3*x* = 2

*x* = 2/3

(*x* + 4) = 0

*x* = – 4

Choice (1).

**Problem 3**

A system of equations is given below.

*x *+ 2*y *= 5

2*x *+ *y =* 4

Which system of equations does *not *have the same solution?

(1) 3*x *+ 6*y *= 15

2*x *+ *y *= 4

(2) 4*x *+ 8*y *= 20

2*x *+ *y *= 4

(3) *x *+ 2*y *= 5

6*x *+ 3*y *= 12

(4) *x *+ 2*y *= 5

4*x *+ 2*y *= 12

A continuación se muestra un sistema de ecuaciones.

* **x *+ 2*y *= 5

2*x *+ *y =* 4

¿Qué sistema de ecuaciones *no *tiene la misma solución?

(1) 3*x *+ 6*y *= 15

2*x *+ *y *= 4

(2) 4*x *+ 8*y *= 20

2*x *+ *y *= 4

(3) *x *+ 2*y *= 5

6*x *+ 3*y *= 12

(4) *x *+ 2*y *= 5

4*x *+ 2*y *= 12

*Discussion and Solution:*

Solution set (1) involves the first equation multiplied by 3, hence they have the same solution.

Solution set (2) involves the first equation multiplied by 4, hence they have the same solution.

Solution set (3) involves the second equation multiplied by 3, hence they have the same solution.

Solution set (4) involves different equation, hence they have different solutions.

Choice (4).

**Problem 4**

Find the zeros of *f*(*x*) = (*x *– 3)^{2} – 49, algebraically.

Encuentre los ceros de *f*(*x*) = (*x –* 3)^{2} – 49, algebraicamente.

*Discussion and Solution:*

In order for the function to be zero, (x – 3)2 must equal 49.

(*x* – 3)^{2} = 49

*x* – 3 = 7

*x* = 10

(*x* – 3)^{2} = 49

*x* – 3 = – 7

*x* = – 4

**Problem 5**

Solve the equation below for *x *in terms of *a*.

4(*ax *+ 3) – 3*ax *= 25 + 3*a*

* *Resuelva la siguiente ecuación para *x *en términos de *a*.

4(*ax *+ 3) – 3*ax *= 25 + 3*a*

*Discussion and Solution:*

First remove the parentheses, then combine like terms:

4(*ax *+ 3) – 3*ax *= 25 + 3*a*

4*ax* + 12 – 3*ax* = 25 + 3*a*

*ax* + 12 = 25 + 3*a*

*ax* = 13 + 3*a*

Divide both sides by *a*:

*ax* = 13 + 3*a*

*x* = (13 + 3a) / a

or

*x* = 13/a + 3