How do you explain yourself in a math answer?

Answer 1

You explain yourself in a math answer by justifying your actions with the use of properties, definitions, theorems and axioms/postulates. This is called a "proof", and represents the foundation of modern mathematics.

For example, here is a simple proof that FOILing two binomials is the same as polynomial multiplication:

Consider the binomials (a*x + b) and (c*x +d)

We know that:

(a*x + b) * (c*x + d) = (a*x + b) * c*x + (a*x + b)*d

by the Distributive Law of Multiplication.

Applying the Distributive Law of Multiplication again a second time, we know that:

(a*x + b) * c*x + (a*x + b) * d = a*x*c*x + b*c*x + a*x*d + b*d.

Applying the Commutative Law of Multiplication and simplifying, we have:

a*x*c*x + b*c*x + a*x*d + b*d = a*c*x^2 + b*c*x + a*d*x + b*d

Applying the Commutative Law of Addition, we have:

a*c*x^2 + b*c*x + a*d*x + b*d = a*c*x^2 + a*d*x + b*c*x + b*D

Since "=" is an equivalence relation, we know the following by transitivity (if h = i and i = j then h = j):

(a*x + b)*(c*x + d) = a*c*x^2 + a*d*x + b*c*x + b*d

That is, the multiplication of two binomials is equal to the summation of the first terms multiplied, the outer terms multiplied, the inner terms multiplied and the last terms multiplied - hence FOIL (First Outer Inner Last).

The above proof is an example of a "direct proof". We showed that two polynomials are equal to each other. THere are other methods for proving: Proof by Cases, Proof by Contradiction, Proof by Contrapositive, Proof by Induction and Proof by Logical Equivalency.