Question

How do you factor \[3{{x}^{2}}+2x-8\] by grouping?

Answer 1

Hint: Apply the middle term split method to factorize \[3{{x}^{2}}+2x-8\]. Split 2x into two parts in such a way that their sum is 2x and the product is \[-24{{x}^{2}}\]. For this process find the prime factors of 24 and combine them in such a way so that we can get our conditions satisfied. Finally, take the common terms together and write \[3{{x}^{2}}+2x-8\] as a product of two terms given as \[\left( x-a \right)\left( x-b \right)\]. Here, ‘a’ and ‘b’ are called zeroes of the polynomial.

Here, we have been asked to factorize the quadratic polynomial: \[3{{x}^{2}}+2x-8\] by grouping.
Let us use the middle term split method for the factorization. It states that we have to split the middle term which is 2x into two terms such that their sum is 2x and product is equal to the product of constant term (-8) and \[3{{x}^{2}}\], i.e., \[-24{{x}^{2}}\]. To do this, first we need to find all the primes factors of 24. So, let us find.
We know that 24 can be written as: - \[24=2\times 2\times 2\times 3\] as the product of its primes. Now, we have to group three 2’s and one 3 such that our condition of the middle term split method is satisfied. So, we have,
(i) \[6x+\left( -4x \right)=2x\]
(ii) \[6x\times \left( -4x \right)=-24{{x}^{2}}\]
Hence, both the conditions of the middle term split method are satisfied. So, the quadratic polynomial can be written as: -
\[\Rightarrow 3{{x}^{2}}+2x-8=3{{x}^{2}}+6x-4x-8\]
Grouping the terms together we have,
\[\Rightarrow 3{{x}^{2}}+2x-8=3x\left( x+2 \right)-4\left( x+2 \right)\]
Taking \[\left( x+2 \right)\] common we have,
\[\Rightarrow 3{{x}^{2}}+2x-8=\left( 3x-4 \right)\left( x+2 \right)\]
Hence, \[\left( 3x-4 \right)\left( x+2 \right)\] is the factored form of the given quadratic polynomial.

Note: One may note that we can use another method for the factorization. The Discriminant method can also be applied to solve the question. What we will do is we will find the solution of the quadratic equation using discriminant method. The values of x obtained will be assumed as x = a and x = b. Finally, we will consider the product \[\left( x-a \right)\left( x-b \right)\] to get the factored form. But you may see that we have been given the question that we must use the grouping method and that is why the middle term split method is used.