# Algebraic Calculations

Maple is an algebraic calculator that allows you to define functions and expressions, factor and expand expressions, convert back and forth from different formats, and find unique algebraic solutions easily.

Let's look at a simple algebraic equation:

```>
200*x^5+60*x^4*y-138*x^3*y^2-47*x^2*y^3+24*x*y^4+9*y^5;

5       4          3  2      2  3         4      5
200 x  + 60 x  y - 138 x  y  - 47x  y  + 24 x y  + 9 y
```

If we wanted to factor this expression, all we would need to do is use the `factor` command:

```> factor(");

2         3
(5 x - 3 y)  (2 x+ y)
```

Likewise, if we wanted to expand this expression we would use the `expand` command:

```> expand(");

5       4          3  2      2  3         4      5
200 x  + 60 x  y - 138 x  y  - 47x  y  + 24 x y  + 9 y
```

It is as simple as that.
Let us go ahead and define a simple expression, sample:

```> sample:=(a^2+bx^2)/(a*b^2*x^2-2*a*b);
2    2
a  +bx
sample :=---------------
2  2
a b  x  -2 a b
```

We used the colon equals `:=` to define the expression sample. Notice that we can `factor` sample:

```> factor(sample);

2     2
a  + bx
--------------
2
a b (b x  -2)
```

Actually, it might be to our advantage to use this factored expression from now on. So we will `factor` sample and save it as newsample:

``` > newsample:=factor(sample);

2    2
a  +bx
newsample :=--------------
2
a b (bx  - 2)
```

Anytime in the future we can use newsample as a true algebraic expression. Suppose we later do some math that involves this expression. We no longer need to type it in:

```> expres:=((b*x^2-2)*newsample);

2    2
a  +bx
expres :=--------
ab

```

Maple already factored the expression. Now, go ahead and assign values for a and b:

```> a:=12.4;

a := 12.4

> b:=-5.2;

b := -5.2

> expres;

2
- 2.384615384 +.08064516127 x
```

Maple can solve equations:

```> myeqn:=5*x^3-3*x^2+x=12.8;

3      2
myeqn := 5 x  - 3 x  +x = 12.8

> solve(myeqn,x);

1.544138995,  - .4720694975 + 1.197928308 I, - .4720694975 - 1.197928308 I
```
Notice that two of the solutions are complex in nature.
Maple can solve multiple equations. First, define the equations:

```> eqn1:=3*x-5.5*y+z = 199.547;

eqn1 := 3 x - 5.5 y + z= 199.547

> eqn2:=9*x-y-5.8*z = -74.634;

eqn2 := 9 x - y - 5.8 z= -74.634

> eqn3:=28*x-15.87*y+43.432*z = 1352.4355;

eqn3 := 28 x - 15.87 y +43.432 z = 1352.4355
```

Next, solve the equations for x, y, and z using the `solve` command. Notice that both the equations and the variables are enclosed in curly brackets, {}:

```> solve({eqn1,eqn2,eqn3},{x,y,z});

{z = 18.99999999, x =.32400000, y = -32.65000000}
```

Expressions are nice, but what about full blown functions? Maple handles them smoothly. So, we will define a function F(x) with maple. It is almost the same as defining an expression. But we will utilize the arrow command ( `->` ) to denote the variables.

```> F := x -> (x^3-4.5*x^2+0.24*x);

3        2
F := x -> x  - 4.5 x + .24 x
```

We can now calculate the value of F for different values of x:

```> F(0);

0

> F(3);

-12.78

> F(34.56);

35911.76602

> F(a+b);

141.696
```

Did you remember that a = 12.4 and b = -5.2 ? Maple did! Now, let's go ahead and define two more functions, G(x) and H(x):

```> G:= x -> log(x);

G := log

> H:= x -> abs(x);

H := abs
```

We can now plot these two functions together:

```> plot( {G,H},-10..10);

AA                                10 +                                  AA
AAA                                 +                                AAA
AA                               +                              AA
AA                           8 +                            AA
AAA                          +                          AA
AAA                        +                       AAA
AAA                      +                     AAA
AA                  6 +                   AA
AAA                  +                 AAA
AAA                +              AAAA
AAA           4 +             AA
AAA           +          AAA
AAA         +        AAA
AA        +      AAA                   BBBBBBBB
AAA   2 +    AAA        BBBBBBBBBBBBBB
AA   +   AA  BBBBBBBB
AA +AAABBBBB
+---+--+---+---+--+---+---+---+--+--***-***-+---+---+---+--+---+---+--+---+
-10               -5                0 0 BB              5                10

```