Question

Let $f(x) = {x^2} + x - 6$.For what values of $t$ is $f(t - 5) = 0$?
$
  {\text{(A) }} - 3{\text{ and }}2 \\
  {\text{(B) }} - 2{\text{ and 3}} \\
  {\text{(C) }}5 \\
  {\text{(D) 2 and 7}} \\
 $

Answer 1

Hint: This is a problem related to factorization of a polynomial function. Factorization is a process to reduce the polynomial function of 2 or higher orders to the factors which cannot be reduced further. These factors can be variables, integers or algebraic expression. Factorization is a reverse process of multiplication.

Complete step-by-step answer:
From the problem, we have been given a function which is a polynomial of 2nd order. This expression can be written as below:
$f(x) = {x^2} + x - 6$
This expression can be factorise in the following way,
$
  f(x) = {x^2} + x - 6 \\
  f(x) = {x^2} + 3x - 2x - 6 \\
  f(x) = x(x + 3) - 2(x + 3) \\
  f(x) = (x + 3)(x - 2) \\
 $
Now, replacing $x$ in the above expression by $t$, we get the function in terms of $t$ as
$f(t) = (t + 3)(t - 2)$
Now, in the above expression, we will replace $t$ with$(t - 5)$and we will get the following expression,
$f(t - 5) = ((t - 5) + 3)((t - 5) - 2)$
After simplification, we get
$f(t - 5) = (t - 2)(t - 7)$
As in the question it is given that$f(t - 5) = 0$, therefore,
$
  0 = (t - 2)(t - 7) \\
  (t - 2)(t - 7) = 0 \\
 $
This way, we get two values of $t$, which are
$t = 2{\text{ and }}7$
Thus, the correct answer is option (D).


Note: Solving problems related to factorization of a polynomial, you must remember the following points while using division method of factorization as:
You have to keep in mind the accuracy of variables and coefficients of the polynomial. A mistake that can lead you to the wrong solution or end you up in a confusing state.
For higher order of polynomial functions, arrange it in the descending order. This will help you to factorise the expression and you can take common factors out of the bracket.