Question

How do you factor \[{{x}^{2}}-23x+132\]?

Answer 1

Hint: Use the middle term split method to factorize \[{{x}^{2}}-23x+132\]. Split -23x into two terms in such a way that their sum equals -23x and product equals \[132{{x}^{2}}\]. For this process, find the prime factors of 132 and combine them in such a manner so that our condition is satisfied. Finally, take the common terms together and write \[{{x}^{2}}-23x+132\] as a product of two terms given as (x – m) (x – n). Here, ‘m’ and ‘n’ are called zeroes of the polynomial.

Complete step by step solution:
Here, we have been asked to factorize the quadratic polynomial: \[{{x}^{2}}-23x+132\].
Let us use the middle term split method for the factorization. It says that we have to split the middle term which is -23x into two terms such that their sum is -23x and product is \[132{{x}^{2}}\]. To do this, first we need to find all the prime factors of 132.
We know that 132 can be written as: - \[132=2\times 2\times 2\times 3\times 5\] as the product of its primes. Now, we have to group these factors such that the conditions of the middle term split method are satisfied. So, we have,
(i) $\left( -15x \right)+\left( -8x \right)=-23x$
(ii) \[\left( -15x \right)\times \left( -8x \right)=132{{x}^{2}}\]
Hence, both the conditions of the middle term split method are satisfied. So, the quadratic polynomial can be written as: -
\[\begin{align}
  & \Rightarrow {{x}^{2}}-23x+132={{x}^{2}}-15x-8x+132 \\
 & \Rightarrow {{x}^{2}}-23x+132=x\left( x-15 \right)-8\left( x-15 \right) \\
\end{align}\]
Taking (x – 15) common in the R.H.S, we have,
\[\Rightarrow {{x}^{2}}-23x+132=\left( x-15 \right)\left( x-8 \right)\]
Hence, \[\left( x-15 \right)\left( x-8 \right)\] is the factored form of the given quadratic polynomial.

Note: Here, we can also use the discriminant method to get the factored form of the given quadratic expression. What we can do is, first we will substitute the given expression equal to 0 and then we will solve the equation by using the quadratic formula: $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ to get the two values of x. Now, we will consider the values of x as x = m and x = n. Now, considering the product (x – m) (x – n) we will get the required factored form.