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To determine if \( x + 5 \) is a factor of the given polynomial:

\[
x^5 - 5x^4 - 9x^3 - 5x^2 + 8x + 5290
\]

we can apply the **Remainder Theorem**. This theorem states that if \( x + 5 \) is a factor of the polynomial, then substituting \( x = -5 \) into the polynomial should give a remainder of zero.

Let's substitute \( x = -5 \) into the polynomial:

\[
P(-5) = (-5)^5 - 5(-5)^4 - 9(-5)^3 - 5(-5)^2 + 8(-5) + 5290
\]

I'll calculate that now.

\[
P(-5) = -3125 - 5(625) - 9(-125) - 5(25) + 8(-5) + 5290
\]

\[
P(-5) = -3125 - 3125 + 1125 - 125 - 40 + 5290
\]

\[
P(-5) = (-3125 - 3125 - 125 - 40) + (1125 + 5290)
\]

\[
P(-5) = -6415 + 6415 = 0
\]

Since the remainder is 0, \( x + 5 \) **is a factor** of the polynomial. The correct answer is that the remainder when divided by \( x + 5 \) is **0**.

Would you like any further details or explanation? 

Here are some related questions to explore:
1. How does synthetic division work for checking factors of polynomials?
2. What is the difference between the Remainder Theorem and the Factor Theorem?
3. How can you factor a polynomial after confirming one factor?
4. What are other methods for factoring polynomials besides using the Remainder Theorem?
5. How would this process change if you were asked to check if \( x - 5 \) is a factor?

**Tip:** Always double-check calculations in long polynomial evaluations to ensure accuracy, especially with large numbers!

Is x + 5 a factor of x^5 - 5x^4 - 9x^3 - 5x^2 + 8x + 5290?

Explore the factorization of the polynomial x^5 - 5x^4 - 9x^3 - 5x^2 + 8x + 5290 using the Remainder Theorem. Learn how to determine if x + 5 is a factor by substituting x = -5 and verifying the remainder. Suitable for students in Grades 10-12.

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