To determine the remainder when the polynomial [tex]\( d^4 + 2d^2 + 5d - 10 \)[/tex] is divided by [tex]\( d + 4 \)[/tex], we can use the Remainder Theorem. The Remainder Theorem states that the remainder of the division of a polynomial [tex]\( f(d) \)[/tex] by [tex]\( d - c \)[/tex] is [tex]\( f(c) \)[/tex].
In this case, we are dividing by [tex]\( d + 4 \)[/tex], which can be written as [tex]\( d - (-4) \)[/tex]. Therefore, according to the Remainder Theorem, we need to evaluate the polynomial [tex]\( f(d) \)[/tex] at [tex]\( d = -4 \)[/tex].
1. Given polynomial: [tex]\( f(d) = d^4 + 2d^2 + 5d - 10 \)[/tex]
2. Substitute [tex]\( d = -4 \)[/tex] into the polynomial:
[tex]\[
f(-4) = (-4)^4 + 2(-4)^2 + 5(-4) - 10
\][/tex]
3. Calculate each term individually:
- [tex]\( (-4)^4 = 256 \)[/tex]
- [tex]\( 2(-4)^2 = 2 \times 16 = 32 \)[/tex]
- [tex]\( 5(-4) = -20 \)[/tex]
- The constant term is [tex]\(-10\)[/tex]
4. Sum these values:
[tex]\[
256 + 32 - 20 - 10 = 258
\][/tex]
Thus, the remainder when [tex]\( d^4 + 2d^2 + 5d - 10 \)[/tex] is divided by [tex]\( d + 4 \)[/tex] is [tex]\(\boxed{258}\)[/tex].
