For the following polynomial​ function, use the remainder theorem to find​ f(k).

f left parenthesis x right parenthesis equals 2 x Superscript 5 Baseline minus 8 x cubed minus 13 x squared minus 25f(x)=2x5−8x3−13x2−25​; 

kequals=33

Question content area bottom

Part 1

​f(33​)equals=enter your response here 

​(Simplify your​ answer.)

To find \( f(33) \) for the polynomial \( f(x) = 2x^5 - 8x^3 - 13x^2 - 25 \) using the **Remainder Theorem**, we proceed as follows:

1. **Apply the Remainder Theorem**: According to the Remainder Theorem, \( f(k) \) is equal to the remainder when \( f(x) \) is divided by \( x - k \).
   
   In this case, we need to calculate \( f(33) \), which means we will substitute \( x = 33 \) directly into \( f(x) \).

2. **Substitute \( x = 33 \) in \( f(x) \)**:
   \[
   f(33) = 2(33)^5 - 8(33)^3 - 13(33)^2 - 25
   \]

3. **Calculate each term individually**:
   - \( 33^5 = 39135393 \)
   - \( 33^3 = 35937 \)
   - \( 33^2 = 1089 \)

   Now substitute these values into the expression:

   \[
   f(33) = 2(39135393) - 8(35937) - 13(1089) - 25
   \]

4. **Simplify each part**:
   - \( 2 \times 39135393 = 78270786 \)
   - \( 8 \times 35937 = 287496 \)
   - \( 13 \times 1089 = 14157 \)

5. **Combine all terms**:
   \[
   f(33) = 78270786 - 287496 - 14157 - 25
   \]

6. **Final Calculation**:
   \[
   f(33) = 77949108
   \]

Therefore, \( f(33) = 77949108 \).

Would you like more details on this process? Here are some related questions:

1. How does the Remainder Theorem help with polynomial evaluations?
2. What steps would be needed if we used synthetic division?
3. How does the result change with different values of \( k \)?
4. Can we estimate large polynomial expressions without expanding fully?
5. Why are terms with zero coefficients omitted in polynomials?

**Tip:** For higher powers of polynomials, breaking down the calculations into smaller steps can reduce errors.

For the following polynomial function, use the remainder theorem to find f(k).

f(x) = 2x^5 - 8x^3 - 13x^2 - 25;

k = 33

f(33) = enter your response here (Simplify your answer.)

Learn how to use the Remainder Theorem to evaluate f(33) for the polynomial function f(x) = 2x^5 - 8x^3 - 13x^2 - 25. This problem covers polynomial concepts, exponentiation, and applying the remainder theorem. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation