Polynomials can be factored using either the synthetic division technique or mid – term splitting (in case of quadratic polynomials). If the quadratic polynomials are possible to be factored, their middle term is split and we get the roots. The other way is to use the formula directly to get two roots. Let us consider some examples to know how polynomial factoring is done:
Example 1: Factorize the polynomial 5 x2 + 8 x + 3 = 0?
Solution: The given polynomial is of degree two and so number of roots it would possess will be 2. As it is possible to split the middle term of this polynomial, we do it as follows:
5x2 + 8 x + 3 = 0,
Or 5 x2 + 5 x + 3 x + 3 = 0,
Or 5 x (x + 1) + 3 (x + 1) = 0,
Or x = -1, -3 /5.
Example 2: Suppose we need to find the highest common factor between x3 + 2x2 + 8 and x2 + x + 4?
Solution: In such a case we use the synthetic division technique as follows:
x3 + 2 x2 + 8 / x2 + x + 4; Remainder = x2 + 4,
x2 + x + 4 / x2 + 4; Remainder = x,
x2 + 4 /x; Remainder = 4,
We see 4 is the common factor between two polynomials x3 + 2 x2 + 8 and x2 + x + 4.
The likelihood ratio test is an approach to match probability of occurrence of a certain value under one theory against the probability of the same value under another theory. The 2nd one is the more limited theory.
These concepts are very important from the perspective of preparing iit entrance exam.