Question

using long division determine which of the following polynomials divides evenlt forna diviser of x+3
A. 4x^3+ 13
B. 6x^7+ -7
C. 12x^9+13
D. 6x^7+ 4x^3

Answer 1

What is the correct answer

Answer 2

To determine which of the given polynomials is divisible by the polynomial \( x + 3 \), we can use the Remainder Theorem. According to this theorem, a polynomial \( f(x) \) is divisible by \( x + c \) if and only if \( f(-c) = 0 \). In this case, we have \( c = 3 \), so we will evaluate each polynomial at \( x = -3 \).

Let's evaluate each polynomial one by one:

**A. \( 4x^3 + 13 \)**
\[
f(-3) = 4(-3)^3 + 13 = 4(-27) + 13 = -108 + 13 = -95 \quad (\text{Not divisible})
\]

**B. \( 6x^7 - 7 \)**
\[
f(-3) = 6(-3)^7 - 7 = 6(-2187) - 7 = -13122 - 7 = -13129 \quad (\text{Not divisible})
\]

**C. \( 12x^9 + 13 \)**
\[
f(-3) = 12(-3)^9 + 13 = 12(-19683) + 13 = -236196 + 13 = -236183 \quad (\text{Not divisible})
\]

**D. \( 6x^7 + 4x^3 \)**
\[
f(-3) = 6(-3)^7 + 4(-3)^3 = 6(-2187) + 4(-27) = -13122 - 108 = -13230 \quad (\text{Not divisible})
\]

After evaluating all the polynomials at \( x = -3 \), none of them resulted in \( 0 \). Thus, none of the polynomials listed (A, B, C, or D) are divisible by \( x + 3 \).

Answer 3

To find out which polynomial is divisible by \( x + 3 \), let's evaluate the polynomials more carefully using polynomial long division. However, since none of the previous evaluations provided a result of zero when substituting \( x = -3 \), I can confirm that none of them are divisible by \( x + 3 \) based on the Remainder Theorem.

### Rechecking with the Remainder Theorem:
1. **A**: \( 4x^3 + 13 \)
- \( f(-3) = 4(-3)^3 + 13 = 4(-27) + 13 = -108 + 13 = -95 \) (not divisible)

2. **B**: \( 6x^7 - 7 \)
- \( f(-3) = 6(-3)^7 - 7 = 6(-2187) - 7 = -13122 - 7 = -13129 \) (not divisible)

3. **C**: \( 12x^9 + 13 \)
- \( f(-3) = 12(-3)^9 + 13 = 12(-19683) + 13 = -236196 + 13 = -236183 \) (not divisible)

4. **D**: \( 6x^7 + 4x^3 \)
- \( f(-3) = 6(-3)^7 + 4(-3)^3 = 6(-2187) + 4(-27) = -13122 - 108 = -13230 \) (not divisible)

### Conclusion
After performing the evaluations correctly, I can confirm once again that none of the polynomials are divisible by \( x + 3 \). Thus, the answer is that there is **no correct answer** among the provided options, as all yield non-zero remainders when evaluated at \( x = -3 \).

If you meant a different polynomial or a different divisor, please clarify, and I'll be happy to assist!