Algebraic calculation : Reminder

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

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.

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. Here is an example of using remarkable identities to factorize an algebraic expression:

• The expression to factor `1+2*x+x^2` is a special expansion type `a^2+b^2+2*a*b`, with `a=1` and `b=x`


• The factorization of `a^2+b^2+2*a*b` is `(a+b)^2`


• By replacing a by `1` and b by `x`, we obtain the factored form of the expression, ie `(1+x)^2`.

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.

• `x+x+3*y-2*y=2*x+y`