In the previous chapter, we have learnt the factorisation of algebraic expressions. But this chapter mainly discusses how to factorise the polynomials. A Polynomial is an expression of the sum of several terms that contain some different powers of an equivalent variable or coefficients. An example of a polynomial with 3 variables is x3 – 5x2yz + yz – 1. For example, polynomials are generally used to form algebraic equations, encode a vast range of problems, from elementary word problems to complicated scientific problems. They are also used to define polynomial functions.
Factorisation of polynomials is mainly the process of breaking a polynomial into the product of its factors. It involves various processes, such as the remainder theorem and the factor theorem.
Solving ax2 + bx + c type equations by first, outside, inside, last to multiply expressions like (x+3) (x+4). Then we have to understand the factoring and leave a space in the brackets like ( ) ( ) for writing the factors. Even we have to leave space for the signs to put. Now we can fill out the first terms and use factoring to guess at the last terms. Then we can test the possibilities for the outside and inside multiplications by narrowing them down to a few possibilities. This was the method that was taught in the previous chapter. The current chapter deals with the factorisation of polynomials with the remainder theorem which states that f(x)= q(x)*g(x) + r(x), where r(x) has a lower degree as compared to g(x).
Later this chapter also deals with the factorisation of trinomials. But if you don't know how to factor for the trinomial ax2+bx+c, then the formula for solving quadratic equations can be used. If a prime number 'p' evenly divides the last two terms and meets some conditions, then the polynomial will be irreducible. The conditions are that the constant term should be a multiple of p but not of p2, and the leading term such as 'a' in ax2 + bx +c should not be the multiple of p.