An expression, in its formal definition, is a statement with all possible combinations of variables, constants and basic algebraic operators. The variables can also have any type of exponent. A term is an expression but in an isolated condition, that is, it is not preceded or followed by any operator. When an expression contains three terms, connected only by addition or subtraction symbols and only when the exponents are non-zero positive integers, then such expressions are called as trinomials.
In many cases trinomials can be modified and can be rewritten in the form of product of two expressions, mostly as product of two binomials. For example, the trinomial x2 + 3xy + 2y2 can be expressed as (x + y)*(x + 2y), which does not at all change the value. This process is called as trinomials factoring. To do this, a good knowledge of factoring techniques is needed. Trinomials factoring helps us in further requirements of a given problem.
As a special case, if a trinomial has only single variable and the power of the variable is 2, then such a trinomial is called as Quadratic. Quadratic expressions and functions are the most found in algebra because many real life situations can be expressed as quadratic functions. When a quadratic function is set equal to 0, then it is termed as quadratic equation. Finding solutions to a quadratic equation is probably the maximum exercise one does in algebra, as it gives the result of many important practical applications. For example, when a ball is thrown in air, the time it hits the ground back is solved by a quadratic equation. In many cases, the solution is arrived by factoring quadratic trinomials and use of zero product property.
Let us take a closer look as to how to factor a quadratic trinomial. In general, a quadratic equation is in the form (or can be made into),
ax2 + bx + c = 0, where a, b and c are constants and a ? 0. The following is the method to factor.
1) Find the product a*c, say =p.
2) List all the possible factors of p.
3) Pick up the set of factors, whose algebraic sum is equal to b.
4) Rewrite the middle term as the sum of the factors found in step 3 and now the trinomial is regrouped.
5) Now it will be possible to factor the regrouped expression as a product of two binomials.
6) Use the zero product property to solve for the variable.
If step 3 does not work out, then the quadratic trinomial cannot be factored. It is called as prime and we need to follow other methods for solution.