Question 4: Factorize:
\((iv)\) \(\; \)\(3x^2+5x+2 \)
Solution:
\(\ \ \ \ 3x^2+5x+2\)
\(= 3x^2+2x+3x+2\)
\(= x(3x+2)+1(3x+2)\)
\( =(3x+2)(x+1)\)
Explanation:
Multiply the coefficient of \(x^2\) (which is \(3\)) with the constant term \(2\):
\(3 \times 2 = 6\)
List the factor pairs of \(6\), and take positive sign with both numbers because middle and constant terms are positive.
Then, Find the sum of each pair and select the one whose sum equals the coefficient of the middle term, which is \(5\).
\((1, 6)\) \(\Rightarrow\) \(\text{sum} = 7\)
\((2, 3)\) \(\Rightarrow\) \(\text{sum} = \mathbf{5} \quad \checkmark\)
So, the correct pair is \((2, 3)\)
Use this pair to split the middle term:
\(3x^2 +5 x + 2\) \(\ =\ \)\(3 x^2 +2x + 3x +2\)
Group the terms and factor: \(= (3x^2 + 2x) + (3x +2)\)
Take common from first two terms and last terms: \(= x(3x +2) + 1(3x + 2)\)
Again, take common from both terms: \(= (3x + 2)(x + 1)\)