Identify The Polynomial Divisor, Dividend, And Quotient Represented By The Synthetic Division.$\[ -3 \ | \ \begin{array}{rrrr} 2 & 11 & 18 & 9 \\ & -6 & -15 & -9 \\ \hline 2 & 5 & 3 & 0 \\ \end{array} \\]Divisor:A. \[$-3\$\] B. 3

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Introduction to Synthetic Division

Synthetic division is a method used in mathematics to divide polynomials. It is a shortcut for polynomial long division and is used to find the quotient and remainder when a polynomial is divided by a linear factor. In this article, we will focus on identifying the polynomial divisor, dividend, and quotient represented by the synthetic division.

What is Synthetic Division?

Synthetic division is a method of dividing polynomials that involves a series of steps. It is used to divide a polynomial by a linear factor of the form (x - a), where a is a constant. The process involves writing down the coefficients of the polynomial in a row, followed by the value of a, and then performing a series of operations to find the quotient and remainder.

The Synthetic Division Process

The synthetic division process involves the following steps:

  1. Write down the coefficients of the polynomial in a row.
  2. Write down the value of a, which is the constant in the linear factor.
  3. Bring down the first coefficient of the polynomial.
  4. Multiply the value of a by the first coefficient and write the result below the second coefficient.
  5. Add the second coefficient and the result from step 4.
  6. Multiply the value of a by the result from step 5 and write the result below the third coefficient.
  7. Add the third coefficient and the result from step 6.
  8. Repeat steps 6 and 7 until all coefficients have been used.
  9. The final result is the quotient and remainder.

Identifying the Polynomial Divisor, Dividend, and Quotient

In the given synthetic division problem, we have the following:

  • Dividend: 2x^3 + 11x^2 + 18x + 9
  • Divisor: x + 3
  • Quotient: 2x^2 + 5x + 3
  • Remainder: 0

Step-by-Step Solution

Let's go through the steps of the synthetic division process to identify the polynomial divisor, dividend, and quotient.

Step 1: Write down the coefficients of the polynomial

The coefficients of the polynomial 2x^3 + 11x^2 + 18x + 9 are 2, 11, 18, and 9.

Step 2: Write down the value of a

The value of a is 3, which is the constant in the linear factor x + 3.

Step 3: Bring down the first coefficient

The first coefficient is 2, so we bring it down.

Step 4: Multiply the value of a by the first coefficient

We multiply 3 by 2 to get 6.

Step 5: Add the second coefficient and the result from step 4

We add 11 and 6 to get 17.

Step 6: Multiply the value of a by the result from step 5

We multiply 3 by 17 to get 51.

Step 7: Add the third coefficient and the result from step 6

We add 18 and 51 to get 69.

Step 8: Multiply the value of a by the result from step 7

We multiply 3 by 69 to get 207.

Step 9: Add the fourth coefficient and the result from step 8

We add 9 and 207 to get 216.

Step 10: Write down the final result

The final result is the quotient and remainder. The quotient is 2x^2 + 5x + 3, and the remainder is 0.

Conclusion

In this article, we have discussed the concept of synthetic division and how to identify the polynomial divisor, dividend, and quotient represented by the synthetic division. We have gone through the steps of the synthetic division process and identified the polynomial divisor, dividend, and quotient in the given problem. Synthetic division is a powerful tool in mathematics that can be used to divide polynomials and find the quotient and remainder.

Frequently Asked Questions

Q: What is synthetic division?

A: Synthetic division is a method used in mathematics to divide polynomials. It is a shortcut for polynomial long division and is used to find the quotient and remainder when a polynomial is divided by a linear factor.

Q: What are the steps of the synthetic division process?

A: The steps of the synthetic division process are:

  1. Write down the coefficients of the polynomial in a row.
  2. Write down the value of a, which is the constant in the linear factor.
  3. Bring down the first coefficient of the polynomial.
  4. Multiply the value of a by the first coefficient and write the result below the second coefficient.
  5. Add the second coefficient and the result from step 4.
  6. Multiply the value of a by the result from step 5 and write the result below the third coefficient.
  7. Add the third coefficient and the result from step 6.
  8. Repeat steps 6 and 7 until all coefficients have been used.
  9. The final result is the quotient and remainder.

Q: How do I identify the polynomial divisor, dividend, and quotient in a synthetic division problem?

A: To identify the polynomial divisor, dividend, and quotient in a synthetic division problem, you need to follow the steps of the synthetic division process. The divisor is the linear factor, the dividend is the polynomial being divided, and the quotient is the result of the division.

References

Glossary

  • Dividend: The polynomial being divided.
  • Divisor: The linear factor by which the polynomial is being divided.
  • Quotient: The result of the division.
  • Remainder: The amount left over after the division.
  • Synthetic Division: A method used in mathematics to divide polynomials.
    Synthetic Division Q&A =========================

Frequently Asked Questions

Q: What is synthetic division?

A: Synthetic division is a method used in mathematics to divide polynomials. It is a shortcut for polynomial long division and is used to find the quotient and remainder when a polynomial is divided by a linear factor.

Q: What are the steps of the synthetic division process?

A: The steps of the synthetic division process are:

  1. Write down the coefficients of the polynomial in a row.
  2. Write down the value of a, which is the constant in the linear factor.
  3. Bring down the first coefficient of the polynomial.
  4. Multiply the value of a by the first coefficient and write the result below the second coefficient.
  5. Add the second coefficient and the result from step 4.
  6. Multiply the value of a by the result from step 5 and write the result below the third coefficient.
  7. Add the third coefficient and the result from step 6.
  8. Repeat steps 6 and 7 until all coefficients have been used.
  9. The final result is the quotient and remainder.

Q: How do I identify the polynomial divisor, dividend, and quotient in a synthetic division problem?

A: To identify the polynomial divisor, dividend, and quotient in a synthetic division problem, you need to follow the steps of the synthetic division process. The divisor is the linear factor, the dividend is the polynomial being divided, and the quotient is the result of the division.

Q: What is the difference between synthetic division and polynomial long division?

A: Synthetic division is a shortcut for polynomial long division. It is used to divide polynomials by a linear factor, while polynomial long division is used to divide polynomials by any factor.

Q: Can I use synthetic division to divide polynomials by a quadratic factor?

A: No, synthetic division is only used to divide polynomials by a linear factor. If you need to divide a polynomial by a quadratic factor, you will need to use polynomial long division.

Q: How do I know if a polynomial can be divided by a linear factor?

A: A polynomial can be divided by a linear factor if it has a root that is equal to the constant in the linear factor.

Q: What is the remainder in synthetic division?

A: The remainder in synthetic division is the amount left over after the division. It is the value of the polynomial at the root of the linear factor.

Q: Can I use synthetic division to find the roots of a polynomial?

A: Yes, synthetic division can be used to find the roots of a polynomial. If the remainder is zero, then the root is a real number. If the remainder is not zero, then the root is a complex number.

Q: How do I use synthetic division to find the roots of a polynomial?

A: To use synthetic division to find the roots of a polynomial, you need to follow the steps of the synthetic division process. If the remainder is zero, then the root is a real number. If the remainder is not zero, then the root is a complex number.

Synthetic Division Examples

Example 1: Dividing a Polynomial by a Linear Factor

Divide the polynomial 2x^3 + 11x^2 + 18x + 9 by the linear factor x + 3.

Solution

Using synthetic division, we get:

  • Dividend: 2x^3 + 11x^2 + 18x + 9
  • Divisor: x + 3
  • Quotient: 2x^2 + 5x + 3
  • Remainder: 0

Example 2: Finding the Roots of a Polynomial

Find the roots of the polynomial x^3 + 2x^2 + 3x + 4.

Solution

Using synthetic division, we get:

  • Dividend: x^3 + 2x^2 + 3x + 4
  • Divisor: x + 1
  • Quotient: x^2 + x + 4
  • Remainder: 0

The root of the polynomial is x = -1.

Synthetic Division Practice Problems

Problem 1: Dividing a Polynomial by a Linear Factor

Divide the polynomial 3x^3 + 5x^2 + 2x + 1 by the linear factor x + 2.

Problem 2: Finding the Roots of a Polynomial

Find the roots of the polynomial x^3 + 3x^2 + 2x + 1.

Synthetic Division Resources

Synthetic Division Glossary

  • Dividend: The polynomial being divided.
  • Divisor: The linear factor by which the polynomial is being divided.
  • Quotient: The result of the division.
  • Remainder: The amount left over after the division.
  • Synthetic Division: A method used in mathematics to divide polynomials.
  • Root: A value of x that makes the polynomial equal to zero.