Select The Correct Answer.One Factor Of The Polynomial 3 X 3 + 20 X 2 − 21 X + 88 3x^3 + 20x^2 - 21x + 88 3 X 3 + 20 X 2 − 21 X + 88 Is ( X + 8 (x + 8 ( X + 8 ]. What Is The Other Factor Of The Polynomial? (Note: Use Long Division)A. ( 3 X 2 − 4 X + 11 (3x^2 - 4x + 11 ( 3 X 2 − 4 X + 11 ] B. ( 3 X − 4 X (3x - 4x ( 3 X − 4 X ] C. $(3x^2

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Introduction

Polynomial equations are a fundamental concept in mathematics, and solving them can be a challenging task. In this article, we will focus on solving polynomial equations using long division, a technique that is essential for factoring polynomials. We will use the given polynomial 3x3+20x2−21x+883x^3 + 20x^2 - 21x + 88 and factor it using long division.

Understanding the Problem

The problem states that one factor of the polynomial 3x3+20x2−21x+883x^3 + 20x^2 - 21x + 88 is (x+8)(x + 8). Our goal is to find the other factor of the polynomial. To do this, we will use long division to divide the polynomial by the given factor.

Step 1: Set Up the Long Division

To set up the long division, we need to divide the polynomial 3x3+20x2−21x+883x^3 + 20x^2 - 21x + 88 by the factor (x+8)(x + 8). We will use the following steps:

  • Divide the leading term of the polynomial (3x33x^3) by the leading term of the factor (xx).
  • Multiply the result by the factor and subtract it from the polynomial.
  • Repeat the process until we have a remainder of zero.

Step 2: Perform the Long Division

Let's perform the long division:

  ____________________
3x^3 + 20x^2 - 21x + 88
-(x + 8)
-------------------
3x^3 + 20x^2 - 21x + 88
- (3x^3 + 24x^2)
-------------------
-4x^2 - 21x + 88
- (-4x^2 - 32x)
-------------------
9x + 88
- (9x + 72)
-------------------
16

Step 3: Find the Other Factor

The result of the long division is 3x2+4x−113x^2 + 4x - 11. This is the other factor of the polynomial. Therefore, the other factor of the polynomial is (3x2+4x−11)(3x^2 + 4x - 11).

Conclusion

In this article, we used long division to factor the polynomial 3x3+20x2−21x+883x^3 + 20x^2 - 21x + 88. We found that the other factor of the polynomial is (3x2+4x−11)(3x^2 + 4x - 11). This technique is essential for solving polynomial equations and is a fundamental concept in mathematics.

Answer

The other factor of the polynomial is (3x2+4x−11)(3x^2 + 4x - 11).

Comparison with Options

Let's compare our result with the given options:

  • Option A: (3x2−4x+11)(3x^2 - 4x + 11)
  • Option B: (3x−4x)(3x - 4x)
  • Option C: (3x2)(3x^2)

Our result matches with option A, but with a slight difference in the sign of the middle term.

Final Answer

Q: What is polynomial division?

A: Polynomial division is a technique used to divide a polynomial by another polynomial. It is a fundamental concept in mathematics and is used to factor polynomials.

Q: How do I perform polynomial division?

A: To perform polynomial division, you need to follow these steps:

  1. Divide the leading term of the polynomial by the leading term of the divisor.
  2. Multiply the result by the divisor and subtract it from the polynomial.
  3. Repeat the process until you have a remainder of zero.

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 use the remainder theorem to find the other factor of a polynomial?

A: To use the remainder theorem to find the other factor of a polynomial, you need to follow these steps:

  1. Divide the polynomial by the given factor.
  2. Use the remainder theorem to find the value of the remainder.
  3. Use the remainder to find the other factor of the polynomial.

Q: What is the difference between polynomial division and factoring?

A: Polynomial division and factoring are two different techniques used to simplify polynomials. Polynomial division is used to divide a polynomial by another polynomial, while factoring is used to express a polynomial as a product of simpler polynomials.

Q: How do I factor a polynomial using polynomial division?

A: To factor a polynomial using polynomial division, you need to follow these steps:

  1. Divide the polynomial by the given factor.
  2. Use the result to find the other factor of the polynomial.

Q: What are some common mistakes to avoid when performing polynomial division?

A: Some common mistakes to avoid when performing polynomial division include:

  • Not following the order of operations.
  • Not multiplying the result by the divisor.
  • Not subtracting the result from the polynomial.
  • Not repeating the process until you have a remainder of zero.

Q: How do I check my work when performing polynomial division?

A: To check your work when performing polynomial division, you need to follow these steps:

  1. Multiply the result by the divisor.
  2. Subtract the result from the polynomial.
  3. Check if the remainder is zero.

Q: What are some real-world applications of polynomial division?

A: Polynomial division has many real-world applications, including:

  • Engineering: Polynomial division is used to design and analyze complex systems.
  • Computer Science: Polynomial division is used to develop algorithms and data structures.
  • Economics: Polynomial division is used to model and analyze economic systems.

Conclusion

In this article, we have answered some frequently asked questions about solving polynomial equations using polynomial division. We have covered topics such as polynomial division, the remainder theorem, factoring, and real-world applications. We hope that this article has been helpful in understanding the concept of polynomial division and its applications.