Question

Using factor theorem factorise \[f\left( x \right) = {x^2} - 5x + 6\]

Answer 1

Hint: Here, we have to use the concept of factorization. Factorization is the process in which a number is written in the form of its small factors which on multiplication give the original number. We will first break the middle term such that we have four terms. We will factor out common terms from the first two terms and the last two terms. Then again factoring out common terms we get the required answer.

Complete step-by-step answer:
Given function is \[f\left( x \right) = {x^2} - 5x + 6\].
In algebra, the factor theorem is a theorem which links the factors and zeros of a polynomial.
Firstly we will split the middle term into two parts such that its multiplication will be equal to the product of the first term and the third term of the equation. Therefore, we get
\[ \Rightarrow f\left( x \right) = {x^2} - 3x - 2x + 6\]
Now we will be taking \[x\] common from the first two terms and taking 2 common from the last two terms. Therefore the equation becomes
\[ \Rightarrow f\left( x \right) = x\left( {x - 3} \right) - 2\left( {x - 3} \right)\]
Now we will take \[\left( {x - 3} \right)\] common from the equation we get
\[ \Rightarrow f\left( x \right) = \left( {x - 3} \right)\left( {x - 2} \right)\]
Hence, (x-3) and (x-2) are the factors of the given function \[f\left( x \right) = {x^2} - 5x + 6\].

Note: Factor theorem is a special case of the polynomial remainder theorem. Remainder Theorem states that if we divide a polynomial by some factor then we will find a smaller polynomial along with a remainder. Other than the factor theorem, we can find factors using synthetic division and polynomial division method. Synthetic division is usually used to find the roots of a polynomial by dividing a polynomial by a linear polynomial. This method is better and simpler than the long division method.