Using long division, determine which of the following polynomials divides evenly for a divisor of x+3 .(1 point) Responses x3−3x2−13x+15 x cubed minus 3 x squared minus 13 x plus 15 x3−5x2+10x−15 x cubed minus 5 x squared plus 10 x minus 15 5x2+7x−12 5 x squared plus 7 x minus 12 3x2−6x+9

Answer 1

To determine which of the given polynomials divides evenly by the divisor \( x + 3 \), we can use polynomial long division or simply evaluate each polynomial at \( x = -3 \). If the result is 0, then the polynomial divides evenly by \( x + 3 \).

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

1. Polynomial: \( x^3 - 3x^2 - 13x + 15 \) \[ (-3)^3 - 3(-3)^2 - 13(-3) + 15 = -27 - 27 + 39 + 15 = 0 \] This polynomial divides evenly by \( x + 3 \).

2. Polynomial: \( x^3 - 5x^2 + 10x - 15 \) \[ (-3)^3 - 5(-3)^2 + 10(-3) - 15 = -27 - 45 - 30 - 15 = -117 \] This polynomial does not divide evenly by \( x + 3 \).

3. Polynomial: \( 5x^2 + 7x - 12 \) \[ 5(-3)^2 + 7(-3) - 12 = 5(9) - 21 - 12 = 45 - 21 - 12 = 12 \] This polynomial does not divide evenly by \( x + 3 \).

4. Polynomial: \( 3x^2 - 6x + 9 \) \[ 3(-3)^2 - 6(-3) + 9 = 3(9) + 18 + 9 = 27 + 18 + 9 = 54 \] This polynomial does not divide evenly by \( x + 3 \).

Based on these evaluations, the polynomial that divides evenly by \( x + 3 \) is \( x^3 - 3x^2 - 13x + 15 \).