Question

Factorize $ 4{x^2} + 9 $ .

Answer 1

Hint: Hint: We can use the formula $ {a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right) $ . This is the difference of square identity and by using this formula we can factorize the given equation. So in order to factorize we have to convert the question and then express it in the form of the difference of square identity.

Complete step by step answer:
Given
 $ 4{x^2} + 9......................\left( i \right) $
Also $ {a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right).....................\left( {ii} \right) $
So we need to express (ii) in terms of (i):
Also we observe that $ 4{x^2} + 9 $ cannot be directly converted to the difference of square identity since it has no linear factors with real coefficients.
So we have to consider complex coefficients to factorize it.
Here on comparing (i) and (ii) we see that instead of a negative sign we have a positive sign. Therefore in order to make the second term negative using two complex relations which are:
 $ i = \sqrt { - 1} \,\;\;\, \Rightarrow {i^2} = - 1\; $
Such that (i) can be written as:
 $ 4{x^2} + 9 = {\left( {2x} \right)^2} - {\left( {3i} \right)^2}..................\left( {iii} \right) $
Now by using (ii) we can write (iii) as below:
 $ {\left( {2x} \right)^2} - {\left( {3i} \right)^2} = \left( {2x + 3i} \right)\left( {2x - 3i} \right) $
Therefore on factorizing $ 4{x^2} + 9 $ we get $ \left( {2x - 3i} \right) $ .
Additional Information:
Another technique for factorizing is grouping, where we take common terms not from the full polynomial but only from the certain terms which have them. So it is used mainly when there is not a common factor for the whole polynomial but there is a common factor only for certain terms.
However here we can’t use grouping since there is no common factor at all.

Note:
While approaching a question one should study it properly and accordingly should choose the method to factorize the polynomial.
Similar questions that have no proper linear factors with real coefficients should be approached using the same method as described above.
Polynomial factorization is always done over some set of numbers which may be integers, real numbers, or complex numbers.