Is x-2 a factor of x5-4⁢x4+3⁢x3+9⁢x2-5⁢x+74? And what would the remainder be?

To determine whether \(x - 2\) is a factor of the polynomial \(f(x) = x^5 - 4x^4 + 3x^3 + 9x^2 - 5x + 74\), we can use the **Remainder Theorem**.

### Remainder Theorem:
- According to the Remainder Theorem, if we divide \(f(x)\) by \(x - 2\), the remainder is \(f(2)\).
- If \(f(2) = 0\), then \(x - 2\) is a factor of \(f(x)\).

Let's compute \(f(2)\):

\[
f(2) = (2)^5 - 4(2)^4 + 3(2)^3 + 9(2)^2 - 5(2) + 74
\]

Calculating each term:

\[
f(2) = 32 - 4(16) + 3(8) + 9(4) - 10 + 74
\]
\[
f(2) = 32 - 64 + 24 + 36 - 10 + 74
\]
\[
f(2) = 32 + 24 + 36 + 74 - 64 - 10
\]
\[
f(2) = 96 - 74
\]
\[
f(2) = 22
\]

### Conclusion:
- Since \(f(2) = 22 \neq 0\), \(x - 2\) is **not** a factor of \(f(x)\).
- The remainder when dividing \(f(x)\) by \(x - 2\) is \(22\).

Would you like further details, or do you have any questions?

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Here are 5 related questions to explore:
1. How would you factor a cubic polynomial?
2. What is the Factor Theorem, and how is it different from the Remainder Theorem?
3. Can you divide the polynomial using synthetic division to verify the remainder?
4. What happens if we divide \(f(x)\) by \(x + 2\) instead?
5. How can you find all factors of a polynomial?

**Tip:** The Remainder Theorem is a great shortcut to quickly check if a linear binomial is a factor of a polynomial!

Is x-2 a factor of x^5 - 4x^4 + 3x^3 + 9x^2 - 5x + 74? And what would the remainder be?

Determine if x - 2 is a factor of the polynomial x^5 - 4x^4 + 3x^3 + 9x^2 - 5x + 74 using the Remainder Theorem. Find the remainder of this division with a detailed, step-by-step solution. Suitable for students in Grades 10-12.

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