Question

Divide x4 + 7 by x - 3.

x³ - 3x² - 9x - 27 R 88
x³ + 3x² + 9x - 27 R -74
x³ + 3x² + 9x + 27 R 88

Answer 1

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

Answer 2

\(x-3\ )\overline{x^4+0x^3+0x^2+0x+7}\)

Answer 3

I think the siplest way is that the remainder = 3^4 +7 = 88
and there is oly one choice left :)

Answer 4

this is remainder theorem or something hehe because the remainder must have less 1degree than divider,so it must be a number without variable