What is the quotient in polynomial form?

Concepts

synthetic division, polynomial division, linear divisor, coefficients

Explanation

This problem asks you to use synthetic division to divide a linear factor into a polynomial. The setup given corresponds to dividing a degree 1 polynomial by (x−2). The coefficients 7 and −18 stand for 7x−18.

Step-By-Step Solution

Step 1

Set up the synthetic division. The number on the left (2) is the value to use for the synthetic division, representing x=2

Coefficients: 7 (for x), −18 (constant)

Carry down the first coefficient:

| 2 | 7 -18

Carry down 7:

| 2 | 7 -18 | | 14 | 7 -4

Step 2

Multiply the carried down 7 by 2:

7×2=14

Write 14 under −18

Step 3

Add −18+14=−4

So the bottom row of synthetic division is 7 and −4

Step 4

Interpret the result:

• The quotient corresponds to 7 (constant, since original polynomial was 7x−18, a degree 1 polynomial)
• −4 is the remainder.

So, x−27x−18​=7, remainder −4

Or, in polynomial form: 7x−18=(x−2)⋅7+(−4)

Quotient in polynomial form: 7 with a remainder of −4

Final Answer

Quotient: 7 Remainder: −4

In polynomial form:

7​
with remainder −4