Wow function
Algebra
Level
3
A certain function f(x) when divided by 2x+3 its remainder is 4. When the same function is divided by 3x+4 its remainder is 5. What is the remainder when you will divided it with 6x^2+17x+12.(hint: the function is a cubic polynomial.)
6x-13
6x+13
13x+6
13x-6
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1 solution
First note that 6 x 2 + 1 7 x + 1 2 = ( 2 x + 3 ) ( 3 x + 4 )
f ( x ) = p ( x ) ( 6 x 2 + 1 7 x + 1 2 ) + y = p ( x ) ( 2 x + 3 ) ( 3 x + 4 ) + y
where y is the remainder.
When we divide f ( x ) by ( 2 x + 3 ) , the term p ( x ) ( 2 x + 3 ) ( 3 x + 4 ) has a factor of ( 2 x + 3 ) in it, so 2 x + 3 y has a remainder of 4, as given in the problem. We can then write y = k 1 ( 2 x + 3 ) + 4
Similarly for ( 3 x + 4 ) , y = k 2 ( 3 x + 4 ) + 5 .
Not we will solve for y. We know y has a degree of 1 on x, so we can let k 1 = 3 and k 2 = 2 so that the coefficients of x match up. From the first equation, we get y = k 1 ( 2 x + 3 ) + 4 = 3 ( 2 x + 3 ) + 4 = 6 x + 1 3
We also plug k 2 = 2 to make sure we get the same result.