Use Synthetic Division To Solve $\left(x^4-1\right) \div (x-1$\]. What Is The Quotient?A. $x^3-x^2+x-1$B. $x^3$C. $x^3+x^2+x+1$D. $x^3-2$

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Introduction

Synthetic division is a method used to divide polynomials by linear factors. It is a powerful tool for simplifying complex polynomial expressions and is widely used in algebra and calculus. In this article, we will use synthetic division to solve the polynomial (x4−1)÷(x−1)\left(x^4-1\right) \div (x-1) and find the quotient.

What is Synthetic Division?

Synthetic division is a method of dividing a polynomial by a linear factor of the form (x−a)(x-a). It is a shortcut method that eliminates the need for long division and is often faster and more efficient than traditional long division. The method involves using a table to perform the division and is based on the concept of the remainder theorem.

How to Perform Synthetic Division

To perform synthetic division, we need to follow these steps:

  1. Write down the coefficients of the polynomial in a row, with the constant term on the right.
  2. Bring down the first coefficient.
  3. Multiply the number at the bottom of the table by the number in the first column and write the result below the next coefficient.
  4. Add the numbers in the second column and write the result below the next coefficient.
  5. Repeat steps 3 and 4 until you reach the last coefficient.
  6. The final number in the bottom row is the remainder.

Solving the Problem

Now that we have a basic understanding of synthetic division, let's use it to solve the problem (x4−1)÷(x−1)\left(x^4-1\right) \div (x-1).

Step 1: Write Down the Coefficients

The polynomial is x4−1x^4-1, which can be written as 1x4+0x3+0x2+0x−11x^4+0x^3+0x^2+0x-1. The coefficients are 1,0,0,0,−11, 0, 0, 0, -1.

Step 2: Perform Synthetic Division

We will use synthetic division to divide the polynomial by (x−1)(x-1). The number in the first column is 11.

1 0 0 0 -1
1 1 0 0 0 -1
1 0 0 0 -1

Step 3: Multiply and Add

We multiply the number at the bottom of the table by the number in the first column and write the result below the next coefficient.

1 0 0 0 -1
1 1 0 0 0 -1
1 0 0 0 -1
1 0 0 0 -1

We add the numbers in the second column and write the result below the next coefficient.

1 0 0 0 -1
1 1 0 0 0 -1
1 0 0 0 -1
1 0 0 0 -1
1 0 0 0 -1

Step 4: Repeat the Process

We repeat the process until we reach the last coefficient.

1 0 0 0 -1
1 1 0 0 0 -1
1 0 0 0 -1
1 0 0 0 -1
1 0 0 0 -1
1 0 0 0 -1

The final number in the bottom row is the remainder.

Step 5: Find the Quotient

The remainder is 11. The quotient is the polynomial without the remainder. In this case, the quotient is x3+x2+x+1x^3+x^2+x+1.

Conclusion

In this article, we used synthetic division to solve the polynomial (x4−1)÷(x−1)\left(x^4-1\right) \div (x-1) and found the quotient to be x3+x2+x+1x^3+x^2+x+1. Synthetic division is a powerful tool for simplifying complex polynomial expressions and is widely used in algebra and calculus. By following the steps outlined in this article, you can use synthetic division to solve a wide range of polynomial division problems.

Answer

The correct answer is C. x3+x2+x+1x^3+x^2+x+1.

References

  • "Synthetic Division" by Math Open Reference
  • "Polynomial Division" by Khan Academy
  • "Synthetic Division" by Purplemath
    Synthetic Division: A Comprehensive Guide =====================================================

Q&A: Frequently Asked Questions

Q: What is synthetic division?

A: Synthetic division is a method used to divide polynomials by linear factors. It is a powerful tool for simplifying complex polynomial expressions and is widely used in algebra and calculus.

Q: How does synthetic division work?

A: Synthetic division involves using a table to perform the division. The method is based on the concept of the remainder theorem. To perform synthetic division, you need to follow these steps:

  1. Write down the coefficients of the polynomial in a row, with the constant term on the right.
  2. Bring down the first coefficient.
  3. Multiply the number at the bottom of the table by the number in the first column and write the result below the next coefficient.
  4. Add the numbers in the second column and write the result below the next coefficient.
  5. Repeat steps 3 and 4 until you reach the last coefficient.
  6. The final number in the bottom row is the remainder.

Q: What are the advantages of synthetic division?

A: Synthetic division has several advantages over traditional long division. It is faster and more efficient, and it eliminates the need for long division. Synthetic division is also a great tool for simplifying complex polynomial expressions.

Q: When should I use synthetic division?

A: You should use synthetic division when you need to divide a polynomial by a linear factor. Synthetic division is a powerful tool for simplifying complex polynomial expressions and is widely used in algebra and calculus.

Q: How do I choose the correct divisor?

A: To choose the correct divisor, you need to identify the linear factor that you want to divide the polynomial by. The divisor should be in the form (x-a), where a is a constant.

Q: What is the remainder theorem?

A: The remainder theorem states that if a polynomial f(x) is divided by (x-a), then the remainder is equal to f(a).

Q: How do I apply the remainder theorem?

A: To apply the remainder theorem, you need to substitute the value of a into the polynomial f(x). The result is the remainder.

Q: What are some common applications of synthetic division?

A: Synthetic division has many applications in algebra and calculus. Some common applications include:

  • Dividing polynomials by linear factors
  • Finding the roots of a polynomial
  • Simplifying complex polynomial expressions
  • Solving systems of equations

Q: How do I practice synthetic division?

A: To practice synthetic division, you can try dividing polynomials by linear factors. You can also use online resources and practice problems to help you improve your skills.

Conclusion

Synthetic division is a powerful tool for simplifying complex polynomial expressions. By following the steps outlined in this article, you can use synthetic division to solve a wide range of polynomial division problems. Remember to choose the correct divisor, apply the remainder theorem, and practice synthetic division to become proficient in this technique.

Additional Resources

  • "Synthetic Division" by Math Open Reference
  • "Polynomial Division" by Khan Academy
  • "Synthetic Division" by Purplemath
  • "Synthetic Division Practice Problems" by IXL

Frequently Asked Questions

  • Q: What is synthetic division? A: Synthetic division is a method used to divide polynomials by linear factors.
  • Q: How does synthetic division work? A: Synthetic division involves using a table to perform the division.
  • Q: What are the advantages of synthetic division? A: Synthetic division is faster and more efficient than traditional long division.
  • Q: When should I use synthetic division? A: You should use synthetic division when you need to divide a polynomial by a linear factor.