Algebra

Example: x − 2 = 4

When we put 6 in place of x we get:

6 − 2 = 4

which is true

So x = 6 is a solution.

How about other values for x ?

• For x=5 we get "5−2=4" which is not true, so x=5 is not a solution.
• For x=9 we get "9−2=4" which is not true, so x=9 is not a solution.
• etc

In this case x = 6 is the only solution.

Example: (x−3)(x−2) = 0

When x is 3 we get:

(3−3)(3−2) = 0 × 1 = 0

which is true

And when x is 2 we get:

(2−3)(2−2) = (−1) × 0 = 0

which is also true

So the solutions are:

x = 3, or x = 2

Example: Solve 3x−6 = 9

Start with:3x−6 = 9 Add 6 to both sides:3x = 9+6 Divide by 3:x = (9+6)/3

Now we have x = something,

and a short calculation reveals that x = 5

Numbers have factors:

And expressions (like x²+4x+3) also have factors:

Example: factor 2y+6

Both 2y and 6 have a common factor of 2:

• 2y is 2 × y
• 6 is 2 × 3

So we can factor the whole expression into:

2y+6 = 2(y+3)

So 2y+6 has been "factored into" 2 and y+3

To simplify an expression, we use like terms.

Look at the expression:

To simplify:

• The terms can be collected together to give .
• The numbers can be collected together to give .

So simplified is

Example:

2 / x−2 + 3 / x+1  =  2 × (x+1) + (x−2) × 3 / (x−2) × (x+1)

And then simplify to:

=   2x+2 + 3x−6 / x²+x−2x−2

=   5x−4 / x²−x−2

Example: Expand 3 × (5+2)

Answer:

It is now expanded.

We can also complete the calculation:

3 × (5+2) = 3 × 5 + 3 × 2
    = 15 + 6
= 21  

Matching up Partners

Two friends (Alice and Betty) challenge 
two other friends (Charles and David) to
individual tennis matches.

How many matches does that make?

• Alice plays Charles, and then Alice plays David
• Then Betty plays Charles and then Betty plays David

They could play in any order, so long as each of the first two friends
gets to play each of the second two friends.

But there is a handy way to help us remember to multiply each term called "FOIL".

It stands for "Firsts, Outers, Inners, Lasts":

• Firsts: ac
• Outers: ad
• Inners: bc
• Lasts: bd

So you multiply the "Firsts" (the first terms of both polynomials), then the "Outers", etc.

Example: (x + 2y)(3x − 4y + 5)

(x + 2y)(3x − 4y + 5)

= 3x² − 4xy + 5x + 6xy − 8y² + 10y

= 3x² + 2xy + 5x − 8y² + 10y

Note: −4xy and 6xy are added because they are Like Terms.

Also note: 6yx means the same thing as 6xy