`f(x) = 3x^(5) +11 x^(4) +90x^(2) - 19x +53` is divided by `x +5` then the remainder is______.

Options

A. 100
B. `-100`
C. `-102`
D. 102

Answer

To find the remainder when the polynomial \( f(x) = 3x^5 + 11x^4 + 90x^2 - 19x + 53 \) is divided by \( x + 5 \), we can use the Remainder Theorem. According to the theorem, the remainder of the division of a polynomial \( f(x) \) by \( x - a \) is equal to \( f(a) \).

### Step-by-Step Solution:

1. **Identify the polynomial and divisor**:
We have the polynomial \( f(x) = 3x^5 + 11x^4 + 90x^2 - 19x + 53 \) and we are dividing by \( x + 5 \).

2. **Rewrite the divisor**:
The divisor \( x + 5 \) can be rewritten as \( x - (-5) \). Thus, we can identify \( a = -5 \).

3. **Apply the Remainder Theorem**:
According to the Remainder Theorem, the remainder \( r \) when dividing \( f(x) \) by \( x + 5 \) is given by \( f(-5) \).

4. **Calculate \( f(-5) \)**:
Substitute \( -5 \) into the polynomial:
\[ f(-5) = 3(-5)^5 + 11(-5)^4 + 90(-5)^2 - 19(-5) + 53 \]

5. **Evaluate each term**:
- Calculate \( 3(-5)^5 = 3 \times (-3125) = -9375 \)
- Calculate \( 11(-5)^4 = 11 \times 625 = 6875 \)
- Calculate \( 90(-5)^2 = 90 \times 25 = 2250 \)
- Calculate \( -19(-5) = 95 \)
- The constant term is \( 53 \)

6. **Combine the results**:
Now, we combine all the calculated values:
\[ f(-5) = -9375 + 6875 + 2250 + 95 + 53 \]

7. **Perform the addition**:
- Start with \( -9375 + 6875 = -2500 \)
- Then, \( -2500 + 2250 = -250 \)
- Next, \( -250 + 95 = -155 \)
- Finally, \( -155 + 53 = -102 \)

8. **Final result**:
Thus, the remainder when \( f(x) \) is divided by \( x + 5 \) is:
\[ \text{Remainder} = -102 \]

### Conclusion:
The remainder when \( f(x) = 3x^5 + 11x^4 + 90x^2 - 19x + 53 \) is divided by \( x + 5 \) is \( \boxed{-102} \).