Who is asking: Student
Level: Secondary
Question:
Given that 4x3 - kx is divisible by 2x+1, find the remainder when the expression is divided by 4 - x2.
By factor theorem, 4*(-1/2)3 - k*(-1/2)=0
Solving k=1
Unfortunately, I cannot go further. Please help!
Hi Raymond,
Substitute k = 1 into 4x3 - kx to get 4x3 - x. Division by the quadratic expression 4 - x2 will leave a remainder which is either a linear expression or a constant.
One method to find this remainder is to use long division, divide by 4 - x2 and record the remainder. A second method is to write 4x3 - x an a multiple of 4 - x2 plus a linear or constant term.
4x3 - x
= x(4x2 - 1)
= 4x(x2 - 1/4)
= 4x(x2 - 4 + 4 - 1/4)
= 4x(x2 - 4) + 4x(4 - 1/4)
= -4x(4 - x2) + 4x( 15/4)
= -4x(4 - x2) + 15x
Thus the remainder, on division by 4 - x2 is 15x.
Penny