For the following polynomial​ function, use the remainder theorem to find​ f(k).f(x)=2x5−10x3−17x2−50​; k=2

To find \( f(k) \) for the given polynomial function \( f(x) = 2x^5 - 10x^3 - 17x^2 - 50 \) using the remainder theorem, we can evaluate \( f(2) \) directly.

The **remainder theorem** states that if a polynomial \( f(x) \) is divided by \( x - k \), the remainder of this division is equal to \( f(k) \).

### Step-by-Step Solution

1. **Substitute \( x = 2 \) into \( f(x) \):**
   \[
   f(2) = 2(2)^5 - 10(2)^3 - 17(2)^2 - 50
   \]

2. **Calculate each term:**
   - \( 2(2)^5 = 2 \times 32 = 64 \)
   - \( -10(2)^3 = -10 \times 8 = -80 \)
   - \( -17(2)^2 = -17 \times 4 = -68 \)
   - \( -50 = -50 \)

3. **Add the results:**
   \[
   f(2) = 64 - 80 - 68 - 50
   \]
   \[
   f(2) = -134
   \]

### Answer
So, \( f(2) = -134 \).

Would you like further explanation or have any questions?

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### Related Questions
1. How do you use the remainder theorem to find remainders of other values?
2. How can synthetic division be used alongside the remainder theorem?
3. What would the answer be if \( k = -1 \)?
4. How can we determine the roots of this polynomial?
5. How does the remainder theorem relate to the factor theorem?

**Tip:** Always check each term separately to avoid calculation errors, especially when substituting values in polynomials with multiple terms.

For the following polynomial function, use the remainder theorem to find f(k). f(x) = 2x^5 - 10x^3 - 17x^2 - 50; k = 2

Explore how to use the remainder theorem to evaluate f(2) for the polynomial function f(x) = 2x^5 - 10x^3 - 17x^2 - 50. This detailed solution covers the remainder theorem concept and polynomial evaluation, suitable for students in Grades 10-12.

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