Final answer:
In order to determine which linear expression divides evenly into 6x²+7x-20, we can use synthetic division. After performing the synthetic division for each given option, we find that none of the linear expressions divide evenly into the given quadratic expression.
Step-by-step explanation:
In order to determine which linear expression divides evenly into 6x²+7x-20, we can use synthetic division. Let's consider each option:
a) (2x-1):
Step 1: Set up the synthetic division:
| 2 -1 -20
Step 2: Bring down the first coefficient:
| 2
Step 3: Multiply the divisor by the value in the bottom row (2) and place it in the second row:
| 2 3
Step 4: Add the values in the second row:
| 2 3 -17
Since the last value in the bottom row is -17, which is not zero, (2x-1) does not divide evenly into 6x²+7x-20.
b) (3x+4):
Step 1: Set up the synthetic division:
| 3 4 -20
Step 2: Bring down the first coefficient:
| 3
Step 3: Multiply the divisor by the value in the bottom row (3) and place it in the second row:
| 3 15
Step 4: Add the values in the second row:
| 3 15 -5
Since the last value in the bottom row is -5, which is not zero, (3x+4) does not divide evenly into 6x²+7x-20.
c) (x-5):
Step 1: Set up the synthetic division:
| 1 -5 -20
Step 2: Bring down the first coefficient:
| 1
Step 3: Multiply the divisor by the value in the bottom row (1) and place it in the second row:
| 1 -4
Step 4: Add the values in the second row:
| 1 -4 -24
Since the last value in the bottom row is -24, which is not zero, (x-5) does not divide evenly into 6x²+7x-20.
d) (2x+5):
Step 1: Set up the synthetic division:
| 2 5 -20
Step 2: Bring down the first coefficient:
| 2
Step 3: Multiply the divisor by the value in the bottom row (2) and place it in the second row:
| 2 14
Step 4: Add the values in the second row:
| 2 14 -6
Since the last value in the bottom row is -6, which is not zero, (2x+5) does not divide evenly into 6x²+7x-20.
None of the given linear expressions divide evenly into 6x²+7x-20.