# Factoring multivariable polynomials

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This solution deals with factoring multivariable polynomials.

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## Step by Step Solution

## Step 1 :

#### Equation at the end of step 1 :

` ((a`^{2}) + 5ab) + (2•3b^{2}) = 0

## Step 2 :

#### Trying to factor a multi variable polynomial :

2.1 Factoring a^{2} + 5ab + 6b^{2}

Try to factor this multi-variable trinomial using trial and error

Found a factorization : (a + 3b)•(a + 2b)

#### Equation at the end of step 2 :

```
(a + 3b) • (a + 2b) = 0
```

## Step 3 :

#### Theory - Roots of a product :

3.1 A product of several terms equals zero.

When a product of two or more terms equals zero, then at least one of the terms must be zero.

We shall now solve each term = 0 separately

In other words, we are going to solve as many equations as there are terms in the product

Any solution of term = 0 solves product = 0 as well.

#### Equation of a Straight Line

3.2 Solve a+3b = 0

Tiger recognizes that we have here an equation of a straight line. Such an equation is usually written y=mx+b ("y=mx+c" in the UK).

"y=mx+b" is the formula of a straight line drawn on Cartesian coordinate system in which "y" is the vertical axis and "x" the horizontal axis.

In this formula :

y tells us how far up the line goes

x tells us how far along

m is the Slope or Gradient i.e. how steep the line is

b is the Y-intercept i.e. where the line crosses the Y axis

The X and Y intercepts and the Slope are called the line properties. We shall now graph the line a+3b = 0 and calculate its properties

#### Graph of a Straight Line :

#### Calculate the Y-Intercept :

Notice that when a = 0 the value of b is 0/3 so this line "cuts" the b axis at b= 0.00000

` b-intercept = 0/3 = 0.00000 `

#### Calculate the X-Intercept :

When b = 0 the value of a is 0/1 Our line therefore "cuts" the a axis at a= 0.00000

` a-intercept = 0/1 = 0.00000 `

#### Calculate the Slope :

Slope is defined as the change in b divided by the change in a. We note that for a=0, the value of b is -0.000 and for a=2.000, the value of b is -0.667. So, for a change of 2.000 in a (The change in a is sometimes referred to as "RUN") we get a change of -0.667 - -0.000 = -0.667 in b. (The change in b is sometimes referred to as "RISE" and the Slope is m = RISE / RUN)

` Slope = -0.667/2.000 = -0.333 `

#### Equation of a Straight Line

3.3 Solve a+2b = 0

Tiger recognizes that we have here an equation of a straight line. Such an equation is usually written y=mx+b ("y=mx+c" in the UK).

"y=mx+b" is the formula of a straight line drawn on Cartesian coordinate system in which "y" is the vertical axis and "x" the horizontal axis.

In this formula :

y tells us how far up the line goes

x tells us how far along

m is the Slope or Gradient i.e. how steep the line is

b is the Y-intercept i.e. where the line crosses the Y axis

The X and Y intercepts and the Slope are called the line properties. We shall now graph the line a+2b = 0 and calculate its properties

#### Graph of a Straight Line :

#### Calculate the Y-Intercept :

Notice that when a = 0 the value of b is 0/2 so this line "cuts" the b axis at b= 0.00000

` b-intercept = 0/2 = 0.00000 `

#### Calculate the X-Intercept :

When b = 0 the value of a is 0/1 Our line therefore "cuts" the a axis at a= 0.00000

` a-intercept = 0/1 = 0.00000 `

#### Calculate the Slope :

Slope is defined as the change in b divided by the change in a. We note that for a=0, the value of b is -0.000 and for a=2.000, the value of b is -1.000. So, for a change of 2.000 in a (The change in a is sometimes referred to as "RUN") we get a change of -1.000 - -0.000 = -1.000 in b. (The change in b is sometimes referred to as "RISE" and the Slope is m = RISE / RUN)

` Slope = -1.000/2.000 = -0.500 `

## Geometric figure: Two Straight Lines

- Slope = -1.000/2.000 = -0.500
- a-intercept = 0/1 = 0.00000
- b-intercept = 0/2 = 0.00000
- Slope = -0.667/2.000 = -0.333
- a-intercept = 0/1 = 0.00000
- b-intercept = 0/3 = 0.00000