Find the value of k so that (x+2) is a factor of 3x³ + kx² + 5x - 27

To find the value of \( k \) such that \( (x+2) \) is a factor of \( 3x^3 + kx^2 + 5x - 27 \), we can use the **Factor Theorem**.

According to the Factor Theorem, if \( (x+2) \) is a factor of the polynomial, then substituting \( x = -2 \) into the polynomial should give 0.

So, substitute \( x = -2 \) into the polynomial \( 3x^3 + kx^2 + 5x - 27 \):

\[
3(-2)^3 + k(-2)^2 + 5(-2) - 27 = 0
\]

Now, simplify each term:

\[
3(-8) + k(4) + (-10) - 27 = 0
\]

\[
-24 + 4k - 10 - 27 = 0
\]

Combine like terms:

\[
-24 - 10 - 27 + 4k = 0
\]

\[
-61 + 4k = 0
\]

Solve for \( k \):

\[
4k = 61
\]

\[
k = \frac{61}{4}
\]

Thus, the value of \( k \) is \( \frac{61}{4} \).

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

Here are 5 related questions for practice:

1. How would you verify that \( (x+3) \) is a factor of a polynomial?
2. What is the Factor Theorem, and how does it relate to polynomial division?
3. How do you find the remainder when dividing a polynomial by \( (x+c) \)?
4. How can synthetic division be used to find the value of \( k \) in similar problems?
5. What other methods can be used to factor polynomials of degree 3?

**Tip**: Always double-check your substitutions and simplifications, especially when working with negative signs and exponents!

Learn how to find the value of k so that (x+2) is a factor of the polynomial 3x³ + kx² + 5x - 27. This problem explores the Factor Theorem and polynomial simplification, suitable for students in Grades 10-12.

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