Question

Factorize \[6{x^2} + 5x - 6\]
A) \[(3x + 2)(2x - 3)\]
B) \[(3x - 2)(2x + 3)\]
C) \[(4x + 3)(3x - 4)\]
D) \[(4x - 3)(3x + 4)\]

Answer 1

Hint:
We will use the method of splitting the middle term to factorize the given polynomial. In order to do this, we will write the middle term as a sum of two terms. Then we will factor out common terms to get the polynomial as a product of two factors.

Complete step by step solution:
The quadratic polynomial given to us is \[6{x^2} + 5x - 6\].
We have to factorize this polynomial i.e.; we have to find factors such that the given polynomial can be expressed as a product of the two factors. To do this, we will use the method of splitting the middle term.
In the given polynomial \[6{x^2} + 5x - 6\], we have to split \[5\] as a sum of two terms whose product is \[6 \times 6 = 36\].
Let us find the factors of 36 and find combinations of numbers that in addition or subtraction will give us \[5\].
We know that \[36 = 2 \times 2 \times 3 \times 3 = 4 \times 9\].
We also know that \[5 = 9 - 4\]. So, the required numbers are 9 and 4. Hence, the polynomial can be written as
\[6{x^2} + 5x - 6 = 6{x^2} + 9x - 4x - 6\]
We will club the first two terms and the last two terms together. In the first two terms the common factor is \[3x\]. In the last two terms, the common factor is \[ - 2\]. Thus,
\[ \Rightarrow 6{x^2} + 5x - 6 = 3x(2x + 3) - 2(2x + 3)\]
We see on the RHS that the factor \[(2x + 3)\] is common to both terms. We will take it out. Hence,
\[ \Rightarrow 6{x^2} + 5x - 6 = (2x + 3)(3x - 2)\]

Therefore, we have expressed the given polynomial as a product of two factors. So, the
correct option is B.

Note:
To factorize a polynomial \[a{x^2} + bx + c\] by splitting the middle term, we have to find two terms such that we can write \[b\] as a sum of the two terms such that their product is \[a \times c\]. This means that we find two numbers \[p\] and \[q\] such that \[p + q = b\] and \[pq = ac\]. After finding \[p\] and \[q\], we split the middle term in the quadratic polynomial as \[px + qx\] and get desired factors by grouping the terms. Here, the terms \[p\] and \[q\] are not necessarily positive numbers.