Question

Factorize: $ {y^2} - 5y + 6 $

Answer 1

Hint: For the equation of the form $ a{x^2} + bx + c = 0 $ , if $ m,n $ are the roots of the equation , then \[m + n = \dfrac{{ - b}}{a}\] and $ m \cdot n = \dfrac{c}{a} $ . Use the substitution method to solve the equations \[m + n = \dfrac{{ - b}}{a}\] and $ m \cdot n = \dfrac{c}{a} $ . After obtaining the values of $ m,{\rm{ }}n $ write it in the form of $ \left( {x - m} \right)\left( {x - n} \right) $ which would be the factorised form of the expression $ a{x^2} + bx + c $ .

Complete step-by-step answer:
First check whether the coefficient of $ {y^2} $ is 1.
The expression $ {y^2} - 5y + 6 $ has 1 as the coefficient of $ {y^2} $ .
This expression is to be written in the form of $ \left( {y - m} \right)\left( {y - n} \right) $ , where $ m.{\rm{ }}n $ are the factors of the equation $ {y^2} - \left( {m + n} \right)y + mn $ .
Now, we should compare $ {y^2} - 5y + 6 $ with $ {y^2} - \left( {m + n} \right)y + mn $ .
On comparing, we get, $ m + n = 5 $ and $ mn = 6 $
If $ m = 2 $ and $ n = 3 $ , then $ 2 + 3 = 5,{\rm{ }}2 \times 3 = 6 $ .
Hence, the expression $ {y^2} - 5y + 6 = \left( {y - m} \right)\left( {y - n} \right) $
Here, substitute the values of $ m = 2 $ and $ n = 3 $ in the equation
$\Rightarrow {y^2} - 5y + 6 = \left( {y - m} \right)\left( {y - n} \right) $
Therefore, $ {y^2} - 5y + 6 $ can be factorised as $ {y^2} - 5y + 6 = \left( {y - 2} \right)\left( {y - 3} \right) $ .

Additional Information:
Factorisation of polynomials is done by separating it in the product of factors which cannot be reduced further. It can also be said as the process of converting addition and subtraction of terms into the form of the product of the terms. Factorisation is necessary to convert the complex looking expressions into the product of linear terms.
 In 1793, Theodor von Schubert produced the first algorithm for the factorization of the polynomial. The linear factors are being deduced from the polynomials in order to factorise the polynomials.

Note: Students often find it difficult to find the factors of the equation. So, it is important to keep in mind that for the equation of the form $ {x^2} - mx + n $ , if the roots are $ a,{\rm{ }}b $ then $ a + b = m $ and $ a \cdot b = n $ .
We need to just compare the given expression to $ {x^2} - mx + n $ and convert it in the form of $ \left( {x - a} \right)\left( {x - b} \right) $ .