develop an algebraic expression with a special expansion

Algebraic calculus is a form of calculus that combines letters, numbers, and operations. The following formulas can be used in factoring and expanding algebraic expressions.

Factoring and expansion

1. Formulas

Let a, b and k be three numbers we have :

• ka+kb=`k*(a+b)`
• ka-kb=`k*(a-b)`
To factor an algebraic sum is to turn it into a product. To expand is to turn it into an algebraic sum.

2. Special expansions

Let a,b be two numbers we have the following three equalities: :

• `(a+b)^2=a^2+2*a*b+b^2`
• `(a-b)^2=a^2-2*a*b+b^2`
• `(a-b)*(a+b)=a^2-b^2`
These equalities are known as special expansions.

3. Factoring by (x-a)

P is a polynomial, a a real. If P(a)=0, then P is factorizable by (x-a). That is, there exists a polynomial Q such that for all x, P(x)=(x-a)Q(x).

Reduction of a literal expression

To simplify a literal expression, we group the dependent terms depending on the same letters and then we reduce each grouping, as in this example: x+x+3y-2y=2x+y.