What is the quotient in polynomial form?

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

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