Using long division, determine which of the following polynomials divides evenly for a divisor of x+3

Question

𝑥
+
3
.(1 point)
Responses

x3−5x2+10x−15
𝑥
3
−
5
𝑥
2
+
10
𝑥
−
15
x cubed minus 5 x squared plus 10 x minus 15

3x2−6x+9
3
𝑥
2
−
6
𝑥
+
9
3 x squared minus 6 x plus 9

5x2+7x−12
5
𝑥
2
+
7
𝑥
−
12
5 x squared plus 7 x minus 12

x3−3x2−13x+15
𝑥
3
−
3
𝑥
2
−
13
𝑥
+
15
x cubed minus 3 x squared minus 13 x plus 15

Answer 1

To determine which of the given polynomials divides evenly by \( x + 3 \), we can use synthetic division. If the remainder is zero, then the divisor divides the polynomial evenly.

In synthetic division, we use the root of the divisor \( x + 3 = 0 \) or \( x = -3 \) to perform the division.

Let's evaluate each polynomial one by one using synthetic division.

1. Polynomial: \( x^3 - 5x^2 + 10x - 15 \)

Using \( -3 \):

-3 |  1  -5   10  -15
   |      -3   24  -90
   ------------------------
     1  -8   34  -105

Remainder: \( -105 \) (not divisible)

2. Polynomial: \( 3x^2 - 6x + 9 \)

Using \( -3 \):

-3 |  3  -6   9
   |     -9  45
   ---------------
     3 -15  54

Remainder: \( 54 \) (not divisible)

3. Polynomial: \( 5x^2 + 7x - 12 \)

Using \( -3 \):

-3 |  5   7   -12
   |     -15   24
   ---------------
     5  -8   12

Remainder: \( 12 \) (not divisible)

4. Polynomial: \( x^3 - 3x^2 - 13x + 15 \)

Using \( -3 \):

-3 |  1  -3  -13  15
   |     -3   18 -45
   ---------------------
     1  -6   5  -30

Remainder: \( -30 \) (not divisible)

Conclusion

None of the given polynomials divide evenly by \( x + 3 \). All of them yield a non-zero remainder.