Problem Solving

kushal.adhia wrote:If r - 5 is a factor of r^2-cr+30 , what is the value of c ?

A. - 6
B. - 3
C. 6
D. 9
E. 11

Please help

Kushal

This question can be solved in two methods.

Method 1: (Methodical one)
(r - 5) is factor (r^2 - cr + 30).
Thus, (r^2 - cr + 30) = (r - 5)*p(r) , where p(r) is linear polynomial in r, i.e. p(r) doesn't contain r^2, r^3... etc. Without loss of generality we can assume p(r) = (r - a), where a is constant.

So, (r^2 - cr + 30) = (r - 5)*(r - a) = r^2 - (a + 5)*r + 5a
Comparing the coefficients of the expressions in LHS and RHS,
5a = 30 and (a + 5) = c

Thus, a = 6 => c = 5 + 6 = 11

The correct answer is E.


Method 1: (Mechanical one)
Try to factorize (r^2 - cr + 30) in such a way that (r - 5) will be factor.

.. (r^2 - cr + 30)
= (r^2 - 5r + 5r - cr + 30)
= [r(r - 5) + (5 - c)r + 30]
= [r(r - 5) + (5 - c)r - 5(5 - c) + 5(5 - c) + 30]
= [r(r - 5) + (5 - c)(r - 5) + 5(5 - c) + 30]
= [(r - 5)(r + 5 - c) + 25 - 5c + 30]
= [(r - 5)(r + 5 - c) + 55 - 5c]

Thus, we see that for (r - 5) to be a factor of (r^2 - cr + 30), (55 - 5c) must be zero.
So, 55 - 5c = 0 => c = 11

The correct answer is E.