Question

Using long division, determine which of the following polynomials divides evenly for a divisor of x+3.(1 point)
Responses

x3−3x2−13x + 15
x3−3x2−13x + 15

x3−5x2+10x − 15
x3−5x2+10x − 15

5x2+7x − 12
5x2+7x − 12

3x2−6x + 9
3x2−6x + 9

Answer 1

To determine if a polynomial divides evenly by \( x + 3 \), we can use synthetic division or long division. However, it's often more straightforward to use synthetic division, especially for linear divisors like \( x + 3 \).

Since we are dividing by \( x + 3 \), we will use the value \( -3 \) for synthetic division.

### 1. Checking \( x^3 - 3x^2 - 13x + 15 \)

Using synthetic division with \( -3 \):

\[
\begin{array}{r|rrrr}
-3 & 1 & -3 & -13 & 15 \\
& & -3 & 18 & -15 \\
\hline
& 1 & -6 & 5 & 0 \\
\end{array}
\]

The remainder is \( 0 \), so \( x^3 - 3x^2 - 13x + 15 \) is divisible by \( x + 3 \).

### 2. Checking \( x^3 - 5x^2 + 10x - 15 \)

Using synthetic division with \( -3 \):

\[
\begin{array}{r|rrrr}
-3 & 1 & -5 & 10 & -15 \\
& & -3 & 24 & -12 \\
\hline
& 1 & -8 & 34 & -27 \\
\end{array}
\]

The remainder is \( -27 \), so \( x^3 - 5x^2 + 10x - 15 \) is not divisible by \( x + 3 \).

### 3. Checking \( 5x^2 + 7x - 12 \)

Using synthetic division with \( -3 \):

\[
\begin{array}{r|rrr}
-3 & 5 & 7 & -12 \\
& & -15 & 24 \\
\hline
& 5 & -8 & 12 \\
\end{array}
\]

The remainder is \( 12 \), so \( 5x^2 + 7x - 12 \) is not divisible by \( x + 3 \).

### 4. Checking \( 3x^2 - 6x + 9 \)

Using synthetic division with \( -3 \):

\[
\begin{array}{r|rrr}
-3 & 3 & -6 & 9 \\
& & -9 & 45 \\
\hline
& 3 & -15 & 54 \\
\end{array}
\]

The remainder is \( 54 \), so \( 3x^2 - 6x + 9 \) is not divisible by \( x + 3 \).

### Conclusion

Among the provided polynomials, the only polynomial that divides evenly by \( x + 3 \) is:

\[
\boxed{x^3 - 3x^2 - 13x + 15}
\]