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When x^3+30x-c is divided by x-5 the remainder is 5. What is c?

When x^4+6x^3-ax^2-45x-15 is divided by x^2-x-6 the remainder is 3. What is a?

Dec 9, 2018

#1
+2

When x^3+30x-c is divided by x-5 the remainder is 5. What is c?

We can use some synthetic division to solve this

5   [  1   0      30       - c   ]

5      25      275

________________

1    5    55       275 - c

So

275 - c = 5     implies that  c = 270   Dec 9, 2018
#2
+2

When x^4+6x^3-ax^2-45x-15 is divided by x^2-x-6 the remainder is 3. What is a?

Haven't done one like this before......!!!!

x^2  + 7x+ (13 - a)

x^2 - x  - 6   [   x^4 + 6x^3 - ax^2    -   45x     -    15 ]

x^4 - x^3 -  6x^2

_____________________________

7x^3 + (6 - a)x^2 - 45x

7x^3 -     7x^2      -42x

___________________________

(13 - a)x^2 - 3x          - 15

(13 - a)x^2 - (13 -a)x - 6(13 - a)

___________________________

(10 - a)x  - 15 + 6(13 - a)

So

-15 + 6(13 - a) = 3

-15 + 78 - 6a = 3

63 - 6a = 3

-6a = -60

a = 10

Check here to see that this is true :

https://www.wolframalpha.com/input/?i=%5B+x%5E4+%2B+6x%5E3+-+10x%5E2+-+45x+%C2%A0-%C2%A0+15%5D+%2F+%5B+x%5E2+-+x+-6%5D   Dec 9, 2018
edited by CPhill  Dec 9, 2018