Use synthetic division to solve what is the quotient

General
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use synthetic division to solve what is the quotient

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Use synthetic division to solve what is the quotient

Synthetic division is a simplified version of polynomial long division, especially useful when dividing by a linear binomial of the form x - c. Here’s a step-by-step guide on how to use synthetic division to find the quotient:

Step-by-Step Process

  1. Set Up the Synthetic Division:

    • Write the coefficients of the polynomial you are dividing.
    • Write the zero of the divisor x - c to the left of the vertical bar.

  2. Bring Down the Leading Coefficient.

    • The first coefficient is brought down as is to the bottom row.

  3. Multiply and Add:

    • Multiply the value you just brought down by the zero of the divisor.
    • Write the result below the next coefficient.
    • Add the values.

  4. Repeat the Process:

    • Repeat steps 2 and 3 for all coefficients.

  5. Interpret the Result:

    • The numbers in the bottom row represent the coefficients of the quotient polynomial.
    • The last number in the bottom row is the remainder.

Example

Let’s use synthetic division to divide 2x^3 + 3x^2 - 4x - 5 by x - 2.

  1. Set Up the Synthetic Division:

    • Coefficients of the polynomial: 2, 3, -4, -5.
    • Zero of the divisor: 2.
    • Set it up as follows:
      2 | 2  3  -4  -5

  2. Bring Down the Leading Coefficient:

    2 | 2  3  -4  -5
------------
  | 2

  3. Multiply and Add:

    • Multiply 2 (left value) by 2 (number just brought down): 2 \times 2 = 4.
    • Write 4 below the second coefficient:
      2 | 2  3  -4  -5

    | 2
    | 4

    - Add $3$ and $4$: $3 + 4 = 7$.
- Continue the process across all coefficients:

    2 | 2 3 -4 -5

    | 2  7   2  -1

  4. Interpret the Result:

    • The bottom row (excluding the remainder) represents the coefficients of the quotient polynomial.
    • The last value is the remainder.

    Thus, the quotient is 2x^2 + 7x + 10 and the remainder is -1.

Final Answer:

The quotient when dividing 2x^3 + 3x^2 - 4x - 5 by x - 2 using synthetic division is 2x^2 + 7x + 10 with a remainder of -1.

Keep practicing synthetic division to become more comfortable with the process and handling different types of polynomial division problems.