Expert: Scott A Wilson - 7/26/2011

Question
31.   Divide:

4u^4-4u^3-5u^2 -9u-1/2u - 1    

32.   Divide:

2a^3 + 8a^2 - 3a + 8/a + 4

33. Divide:

2x^4 - x^3 + x^2 + 4x - 4/2x^2 + x - 2

34.   Find f(c) for the given polynomial f(x) and value c:

f(x) = 3x3 + 8x2 - 6x + 4; c = -4

35.   Find the remaining roots for the following equation:

2x3 - x2 + 10x - 5 = 0; iÖ   5

Answer
To divide, use sythetic division
31. Divide 4 -4 -5 -9 -1 by 2 -1
2 goes into 4 twice, so muliply [2,-1] by 2 and get [4,-2].
Subtracting this from the first two terms, [4,-4], gives [0,-2].
Bring down the -2 and then add the -5, so you've got [-2,-5].
To divide that by [2,-1], multiply by -1 and subtract.
Taking [2,-1] by -1 gives [-2,1], and subtracting that from [-2,-5] gives [0,-6].
The next term is -9, so we have [-6,-9].
Note -6/2 is -3, so multiply [2,-1] by -3, giving [-6,3] and subtract from [-6,-9].
This gives [0,-12], and again we drop the 0 and add the next term, givien [-12,-1].
Dividing -12 by 2 gives -6, so multiply [2,-1] by -6 and get [-12,6].
Subtracting [-12,6] from [-12,-1] gives [0,-7], so the remainder is -7.

To say the answer is u³ - u² + 3u - 6, remainder -7

32. Do division in a similar fashion and I get 2a² + 3, remainder -20;
multiply by a+4 to see if correct.

33. I get x² - x + 2; here, the divisor is [2,1,-2].

34. Rewrite the function as f(x) = ((3x + 8)x - 6)x + 4.
Putting in -4 for x gives ((3(-4) + 8)(-4) - 6)(-4) + 4.
Now 3*-4 is -12, and -12 + 8 is -4, so we have (-4(-4) - 6)(-4) + 4.
Since -4 times -4 is 16, and 16 - 6 is 10, we have 10(-4) + 4.
Now 10 times -4 is -40, and +4 gives us -36.

35. Factoring this gives (2x-1)(x²+5).
This means the roots are found by solving 2x-1 = 0 and x²+5 = 0.
This gives 2x=1, so x=1/2 and x²=-5, so x=±squareoot(5)*i, where i²=-1 {or i = squareroot(-1)}.