To find the remainder when the polynomial x4 + 36 is divided by x2 + 8, we can use polynomial long division or synthetic division. However, for many polynomial operations, a more straightforward approach is to express the dividend in a suitable form.
1. **Expressing the Dividend:** Start with:
x4 + 36
2. **Dividing Terms**: We know that x4 can be factored in terms of x2, as follows:
x4 = (x2)2
3. **Setting Up the Expression:** We want to rewrite x4 + 36 so we can separate it based on the divisor x2 + 8:
x4 + 36 = (x2 + 8)(x2 + A) + R
where A is some constant to be determined, and R will be the remainder we’re looking for.
4. **Polynomial Long Division:** Performing the division directly:
- Divide the leading term of x4 by the leading term of x2, giving x2.
- Multiply x2 by x2 + 8: x4 + 8x2.
- Subtract this from x4 + 36:
(x4 + 36) – (x4 + 8x2 ) = 36 – 8x2
5. **Divide Again:** Now, take the result 36 – 8x2 and continue:
- The highest degree term is now -8x2.
- Divide -8x2 by x2, which gives -8.
- Multiply -8 by x2 + 8:
Resulting in: -8x2 – 64
6. **Final Subtraction:** Subtract again:
(36 – 8x2}) – (-8x2 – 64) = 100
Thus, after performing this polynomial long division, we find that:
Remainder: 100
In summary, the remainder when x4 + 36 is divided by x2 + 8 is 100.