Divide x4 + 7 by x - 3 ... A] x³ - 3x² - 9x - 27 R 88. B] x³ + 3x² + 9x - 27 R - 74. C] x³ + 3x² + 9x + 27 R 88

divide a polynomial p (x) by (x-3). Add and subtract the multiple of (x-3) that has the same highest-power term as p (x), then simplify to get a smaller-degree polynomial r (x) plus multiple of (x-3).

The multiple of (x-3) that has x^4 as its leading term is x^3 (x-3) = x^4 - 3x^3. So write:

x^4 + 7 = x^4 + 7 + x^3 (x - 3) - x^3 (x - 3)

= x^4 + 7 + x^3 (x - 3) - x^4 + 3x^3

= x^3 (x - 3) + 3x^3 + 7

That makes r (x) = 3x^3 + 7. Do the same thing to reduce r (x) by adding/subtracting 3x^2 (x - 3) = 3x^3 - 9x^2:

= x^3 (x - 3) + 3x^3 + 7 + 3x^2 (x - 3) - (3x^3 - 9x^2)

= x^3 (x - 3) + 3x^2 (x - 3) + 9x^2 + 7

Again to reduce 9x^2 + 7:

= x^3 (x - 3) + 3x^2 (x - 3) + 9x^2 + 7 + 9x (x - 3) - (9x^2 - 27x)

= x^3 (x - 3) + 3x^2 (x - 3) + 9x (x - 3) + 27x + 7

And finally write 27x + 7 as 27 (x - 3) + 88;

x^4 + 7 = x^3 (x - 3) + 3x^2 (x - 3) + 9x (x - 3) + 27 (x - 3) + 88

Factor out (x - 3) in all but the + 88 term:

x^4 + 7 = (x - 3) (x^3 + 3x^2 + 9x + 27) + 88

That means that:

(x^4 + 7) / (x - 3) = x^3 + 3x^2 + 9x + 27 with a remainder of 88