If r  5 is a factor of r^2cr+30 , what is the value of c ?
A.  6
B.  3
C. 6
D. 9
E. 11
Please help
Kushal
quad eqn factors
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This question can be solved in two methods.kushal.adhia wrote:If r  5 is a factor of r^2cr+30 , what is the value of c ?
A.  6
B.  3
C. 6
D. 9
E. 11
Please help
Kushal
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.
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 waltz2salsa
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Another way:
r^2  cr + 30; let r1, r2 be factors
sum of roots, r1+r2 = c
product of roots, r1*r2 = 30
if r1=5, solve to get r2 = 6
hence c= 11
Regards,
Shashwat
r^2  cr + 30; let r1, r2 be factors
sum of roots, r1+r2 = c
product of roots, r1*r2 = 30
if r1=5, solve to get r2 = 6
hence c= 11
Regards,
Shashwat