https://file.aiquickdraw.com/mathbot/file/1726516989764xauyoiwr.jpg

The image contains several polynomial expressions, and you are asking for factorization using the Factor Theorem. The Factor Theorem states that if \( f(a) = 0 \), then \( x - a \) is a factor of the polynomial \( f(x) \).

Here are the polynomials and their corresponding divisors from the image:

1. \( (y + 3) \) for \( 2y^3 + y^2 - 13y + 6 \)
2. \( (b - 2) \) for \( 4b^3 - 3b^2 - 8b + 4 \)
3. \( (x + 3) \) for \( x^3 + 2x^2 - 5x - 6 \)
4. \( (x + 1) \) for \( x^{20} - 4x^{18} + 3x - 6 \)
5. \( (x + 1) \) for \( 6x^3 + 13x^2 + 9x + 2 \)

Let's go step by step and check whether the given binomials are factors using the Factor Theorem:

### 1. \( (y + 3) \) for \( 2y^3 + y^2 - 13y + 6 \)
We substitute \( y = -3 \) into the polynomial \( 2y^3 + y^2 - 13y + 6 \):
\[
f(-3) = 2(-3)^3 + (-3)^2 - 13(-3) + 6 = 2(-27) + 9 + 39 + 6 = -54 + 9 + 39 + 6 = 0
\]
Since \( f(-3) = 0 \), \( (y + 3) \) is a factor.

### 2. \( (b - 2) \) for \( 4b^3 - 3b^2 - 8b + 4 \)
We substitute \( b = 2 \) into the polynomial \( 4b^3 - 3b^2 - 8b + 4 \):
\[
f(2) = 4(2)^3 - 3(2)^2 - 8(2) + 4 = 4(8) - 3(4) - 16 + 4 = 32 - 12 - 16 + 4 = 8
\]
Since \( f(2) \neq 0 \), \( (b - 2) \) is **not** a factor.

### 3. \( (x + 3) \) for \( x^3 + 2x^2 - 5x - 6 \)
We substitute \( x = -3 \) into the polynomial \( x^3 + 2x^2 - 5x - 6 \):
\[
f(-3) = (-3)^3 + 2(-3)^2 - 5(-3) - 6 = -27 + 2(9) + 15 - 6 = -27 + 18 + 15 - 6 = 0
\]
Since \( f(-3) = 0 \), \( (x + 3) \) is a factor.

### 4. \( (x + 1) \) for \( x^{20} - 4x^{18} + 3x - 6 \)
We substitute \( x = -1 \) into the polynomial \( x^{20} - 4x^{18} + 3x - 6 \):
\[
f(-1) = (-1)^{20} - 4(-1)^{18} + 3(-1) - 6 = 1 - 4 + (-3) - 6 = -12
\]
Since \( f(-1) \neq 0 \), \( (x + 1) \) is **not** a factor.

### 5. \( (x + 1) \) for \( 6x^3 + 13x^2 + 9x + 2 \)
We substitute \( x = -1 \) into the polynomial \( 6x^3 + 13x^2 + 9x + 2 \):
\[
f(-1) = 6(-1)^3 + 13(-1)^2 + 9(-1) + 2 = 6(-1) + 13(1) + 9(-1) + 2 = -6 + 13 - 9 + 2 = 0
\]
Since \( f(-1) = 0 \), \( (x + 1) \) is a factor.

### Summary
- \( (y + 3) \) is a factor of \( 2y^3 + y^2 - 13y + 6 \)
- \( (b - 2) \) is **not** a factor of \( 4b^3 - 3b^2 - 8b + 4 \)
- \( (x + 3) \) is a factor of \( x^3 + 2x^2 - 5x - 6 \)
- \( (x + 1) \) is **not** a factor of \( x^{20} - 4x^{18} + 3x - 6 \)
- \( (x + 1) \) is a factor of \( 6x^3 + 13x^2 + 9x + 2 \)

Would you like me to provide more details or proceed with any further steps?

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**Relative Questions:**
1. How do we factor higher-degree polynomials when the Factor Theorem doesn't apply?
2. Can we use synthetic division to check factors?
3. What is the difference between the Factor Theorem and the Remainder Theorem?
4. How can we factor polynomials of degree greater than three?
5. How do we apply the Rational Root Theorem in factoring polynomials?

**Tip:** When checking for factors of a polynomial, synthetic division is a quick and efficient alternative to substituting values directly.

Factor the polynomials using the Factor Theorem.

Learn how to factor polynomials using the Factor Theorem, including examples such as 2y^3 + y^2 - 13y + 6, 4b^3 - 3b^2 - 8b + 4, and more. Step-by-step solutions are provided, demonstrating the application of the Factor Theorem and the identification of factors.

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