# Algebra II Hints

This section consists of Algebra II hints, based on common errors that I see on student tests. I also provide a helpful hints for Algebra I, which would also be helpful.

## Algebra I

These are the topics in the Algebra I page.

## Algebra II

As common problems come up in Algebra II, I will add to this section as needed. The topics are as follows:

### Combining Function

Functions come in the general format of f(x) = function of x. I.E. f(x) = x2 -3x + 2. Functions can be combined or act on one another. There are specific rules for combining functions. Following are a few basic examples.

Function How to Use
Functions f(x)=2x2+4x-8, g(x)= 5x-3, h(x)=1/(x-2)
(f+g)(x) f(x)+g(x) = (2x2+4x-8) + (5x-3) = 2x2+4x-8+5x-3 = 2x2+9x-11
(f+g)(-2) f(-2)+g(-2) = (2(-2)2+4(-2)-8)+(5(-2)-3) = 2(-2)2+4(-2)-8+5(-2)-3 = 2(4)+9(-2)-11=8-18-11=-21
(f-g)(x) f(x)-g(x) = (2x2+4x-8) – (5x-3) = 2x2+4x-8-5x+3 = 2x2-x-5
(f-g)(-2) f(-2)-g(-2) = (2(-2)2+4(-2)-8)-(5(-2)-3) = 2(-2)2+4(-2)-8-5(-2)+3 = 2(4)-(-2)-5=8+2-5=5
(f.g)(x) f(x).g(x) = (2x2+4x-8).(5x-3) = 10x3+20x2-40x-6x2-12x+24
= 10x3+14x2-52x+24
(f.g)(-2) f(-2).g(-2) =
(f/g)(x) f(x)/g(x) = (2x2+4x-8)/(5x-3)
(f/g)(-2) f(-2)/g(-2) = (2(-2)2+4(-2)-8)/(5(-2)-3) = (8+4)/(-10-3) = 12/(-13) = -12/13
(goh)(x) (g(h(x)) = g(1/(x-2)) = 5(1/(x-2)) – 3 = 5/(x-2) – 3(x-2)/(x-2)
= 5/(x – 2) – (3x – 6)/(x-2) = (5 – 3x + 6)/(x -2) = (-3x + 11)/(x-2)
(goh)(-2) (g(h(x)) = g(1/(-2-2)) = 5(1/(-4)) – 3 = 5/(-4) – 3(-4)/(-4)
= 5/(-4) – (-12)/(-4) = (5 + 12)/(-4) = (17)/(-4) = -4 1/4
(hog)(x) (h(g(x))

### FOIL

In multiplying expressions, every element of the first expression must be multiplied by every element of the second expression. Normally, this is a two by two multiplication and the way to remember how this is done is through FOILFirst, Outer, Inner, and Last. For example:

```(ax + b)(cx + d) = acx2+ adx + bcx + bd = acx2 + (ad + bc)x + bd

In most cases a and c are 1, so the middle expression becomes (b + d)x. In this case, the number in front of the x is the sum of the two number that when multiplied gives the final element of the resulting equation. For example:```
```(x + 3)(x + 5) = x2 + 5x+3x + 15 = x2 + (5+3)x + 15 = x2 + 8x + 15
(x + 3)(x - 5) = x2 - 5x+3x - 15 = x2 + (-5+3)x - 15 = x2 - 2x - 15
(x - 3)(x + 5) = x2 + 5x-3x - 15 = x2 + (5-3)x - 15 = x2 + 2x - 15
(x - 3)(x - 5) = x2 - 5x-3x + 15 = x2 - (5+3)x + 15 = x2 - 8x + 15
```

### Factoring

Use FOIL in reverse to calculate the factors from a polynomial.

### Rational Expressions

A Rational Expression is a ratio, or fraction, that includes variables in the elements of the ratio. For example:

``` x - 3    =          x - 3         =        1
x2-9              (x-3)(x+3)            (x+3)
```

#### Simplifying

Simplifying is finding common factors, like in the example above, and cancelling them out.

#### Restricted Values

Restricted Values are values not possible for the value of the variable. In the example above, you would set the denominator to 0 and then solve for X. The two possible solutions are given for before and after simplification:

```Before: (x-3)(x+3)=0->(x-3)=0 or x=+3 and (x+3) = 0 or x=-3
After:  (x+3) = 0 or x = -3
```

It is better to simplify first and then find the domain from -∞ through +∞, with the exception for the Restricted Values.

```Before: (-∞, -3) ∪ (-3, 3) ∪ (3, ∞)
After:  (-∞, -3) ∪ (-3, ∞)
```

### Two Variable set of equations

As a rule of thumb, you need as many equations as you do variables in order to solve a problem. In the examples here, there will be two unknowns so we need to create two equations and solve for them.

#### Two person bike ride

If Josh and Janice go for a bike ride. They decide on a 24-mile route. Janice rides 2 miles/hour faster than Josh and finishes the course 1 hour sooner. What are their speeds and times. The variables are:

Time = Distance/Speed

Rider Speed Time Time2 = Distance/Speed
Josh x mph y hours 24 miles/x mph
Janice x+2 mph y-1 hours 24 miles/((x+2) mph)

Distance = speed * time. Therefore, plugging in for y from the Josh line into the Janice equation, I have:

```((24 hours)/(x mph) -1 hour)((x + 2) mph) = 24 miles
Multiplying both sides by x produces:
(24 - 1x)(x+2) =24x
Using FOIL:
24x + 48 - x2 - 2x = 24x
Combining:
-x2 + 22x + 48 = 24x
+x2 - 22x - 48 = +x2 - 22x - 48
0 = x2 +2x -48
Factoring:
(x + 8)(x - 6) = 0
Since speed cannot be a negative number, the only answer is:
x = 6 mph
y = 24 miles/6 mph = 4 hours
```

Then the final table is:

Rider Speed Time Time2 = Distance/Speed
Josh 6 mph 4 hours 24 miles/6 mph
Janice 8 mph 3 hours 24 miles/8 mph