given that 4x^4 -12x^3 - b^2 x^2 - 7bx - 2 is exactly divisible by 2x + b, (i) show that 3b^3 + 7b^2 - 4 = 0 (ii) find the possible values of b

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given that 4x^4 -12x^3 - b^2 x^2 - 7bx - 2 is exactly divisible by 2x + b, (i) show that 3b^3 + 7b^2 - 4 = 0 (ii) find the possible values of b

Mathematics
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I gtg - but will check up on this later :)
Have you been studying polynomial division or synthetic division? If we use synthetic division, see the attached image file and multiply the result by 2 to clear the fractions we can see the remainder is is equal to 3b^3 + 7b^2 - 4. Since they are telling us 4x^4 -12x^3 - b^2 x^2 - 7bx - 2 is exactly divisible by 2x + b, then we can set the remainder = 0. Now we have to solve this. It would be nice if it were a quadratic because then we could just factor it, or use the quadratic formula.. Since it is degree 3, we need to do something else. We can use the ration root theorem p/q idea If you use x = -1, then we get a remainder of 0, which tells us x = -1 See division2.jpg This leaves us 3x^2+4x-4 = 0 This factors to (x+2)(3x-2)=0 which gives us x = -2 and x = 2/3 and the earlier zero of x = 1 Does that make sense?
<3 I figured it out before xD thx!

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