Question

Using long division, determine which of the following polynomials divides evenly for a divisor of  x+3 .(1 point)Responsesx3−5x2+10x−15x cubed minus 5 x squared plus 10 x minus 153x2−6x+93 x squared minus 6 x plus 95x2+7x−125 x squared plus 7 x minus 12x3−3x2−13x+15

Answer 1

To determine which polynomial divides evenly by \( x + 3 \), we can use polynomial long division or just evaluate the polynomials at \( x = -3 \). If a polynomial \( f(x) \) is divisible by \( x + 3 \), then \( f(-3) = 0 \).

Let's evaluate each polynomial at \( x = -3 \):

1. **For \( f(x) = x^3 - 5x^2 + 10x - 15 \)**:
\[
f(-3) = (-3)^3 - 5(-3)^2 + 10(-3) - 15 = -27 - 45 - 30 - 15 = -117 \quad (\text{not divisible})
\]

2. **For \( g(x) = 3x^2 - 6x + 9 \)**:
\[
g(-3) = 3(-3)^2 - 6(-3) + 9 = 3(9) + 18 + 9 = 27 + 18 + 9 = 54 \quad (\text{not divisible})
\]

3. **For \( h(x) = 5x^2 + 7x - 12 \)**:
\[
h(-3) = 5(-3)^2 + 7(-3) - 12 = 5(9) - 21 - 12 = 45 - 21 - 12 = 12 \quad (\text{not divisible})
\]

4. **For \( k(x) = x^3 - 3x^2 - 13x + 15 \)**:
\[
k(-3) = (-3)^3 - 3(-3)^2 - 13(-3) + 15 = -27 - 27 + 39 + 15 = 0 \quad (\text{divisible})
\]

After evaluating all the polynomials, we find that the polynomial \( x^3 - 3x^2 - 13x + 15 \) divides evenly by \( x + 3 \).

Thus, the correct response is:

**\( x^3 - 3x^2 - 13x + 15 \)**