Question

Factorize the given quadratic polynomial: - \[{{x}^{2}}-2x-8\].

Answer 1

Hint: Use the middle term split method to factorize, \[{{x}^{2}}-2x-8\]. Split -2x into two terms in such a way that their sum is -2x and the product is \[-8{{x}^{2}}\]. For this process, find the prime factors of 8 and combine them in such a manner so that we can get our condition satisfied. Finally take the common terms together and write \[{{x}^{2}}-2x-8\] 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.

Complete step-by-step solution:
Here, we have been asked to factorize the quadratic polynomial, \[{{x}^{2}}-2x-8\].
Let us use the middle term split method for the factorization. It says that we have to split the middle term which is \[-2x\], into two terms, such that their sum is \[-2x\] and the product is \[-8{{x}^{2}}\]. To do this first, we need to find all the prime factors of 8. So, let us find it.
We know that 8 can be written as: - \[8=2\times 2\times 2\], as the product of its primes. Now, we have to group these three 2’s such that our condition of the middle term split method is satisfied.
Let us consider 4 and 2 such that after combining with ‘x’ we get -4x and 2x. Clearly, we can see that,
\[\begin{align}
  & \Rightarrow -4x+2x=-2x \\
 & \Rightarrow \left( -4x \right)\times \left( 2x \right)=-8{{x}^{2}} \\
\end{align}\]
Hence, both the conditions of the middle term split method are satisfied. So, the quadratic polynomial can be written as: -
\[\Rightarrow {{x}^{2}}-2x-8={{x}^{2}}-4x+2x-8\]
Grouping the terms together, we have,
\[\begin{align}
  & \Rightarrow {{x}^{2}}-2x-8=\left( {{x}^{2}}-4x \right)+\left( 2x-8 \right) \\
 & \Rightarrow {{x}^{2}}-2x-8=x\left( x-4 \right)+2\left( x-4 \right) \\
\end{align}\]
Taking (x – 4) common, we get,
\[\Rightarrow {{x}^{2}}-2x-8=\left( x+2 \right)\left( x-4 \right)\]
Hence, \[\left( x+2 \right)\left( x-4 \right)\] is the factorized form of the original 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 to the quadratic equation using the 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 factorized form.