Question

Without actual division prove that (x2

- x - 2) divides 2x4 + x3

- 5x2

- 8x – 4​

Answer 1

Step-by-step explanation:

If x²-x-2 divides 2x⁴+x³-5x²-8x-4then the roots of x²-x-2 will satisfy the given polynomial.

so first we will find the roots of x²-x-2 .By splitting the middle term.

x²-x-2 =0

x²-2x +x-2=0

x(x-2)+1(x-2)=0

(x+1)(x-2)=0

x=2,-1.

Now let's check by putting x=2 in 2x⁴+x³-5x²-8x-4

2×2⁴+2³-5×2²-8×2-4

32+8-20-16-4=0

Hence x²-x-2divides 2x⁴+x³-5x²-8x-4

Answer 2

x²-x-2 =0

x²-2x +x-2=0

x(x-2)+1(x-2)=0

(x+1)(x-2)=0

x=2,-1.

putting (x=2) in the polynomial

2×2⁴+2³-5×2²-8×2-4

32+8-20-16-4=0

(x²-x-2)divides (2x⁴+x³-5x²-8x-4)