Use Synthetic Division To Find All The Factors Of This Polynomial:$ 4x^3 + 5x^2 - 18x + 9 }$Possible Factors To Consider - { 2x - 3 $ $- { X + 3 $}$- { 4x - 3 $}$- { X - 1 $} − \[ - \[ − \[ X - 3

Introduction

Synthetic division is a method used to divide polynomials by linear factors. It is a powerful tool for factoring polynomials and is often used in algebra and calculus. In this article, we will use synthetic division to find all the factors of the polynomial $4x^3 + 5x^2 - 18x + 9$. We will consider five possible factors: $2x - 3$, $x + 3$, $4x - 3$, $x - 1$, and $x - 3$.

What is Synthetic Division?

Synthetic division is a method of dividing polynomials by linear factors. It is a shortcut method that eliminates the need for long division. The method involves using a single number, called the divisor, to divide the polynomial. The divisor is usually a linear factor, such as $x - a$, where $a$ is a constant.

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. Write down the divisor, which is usually a linear factor.
3. Bring down the first coefficient of the polynomial.
4. Multiply the divisor by the first coefficient and write the result below the second coefficient.
5. Add the second coefficient and the result from step 4.
6. Repeat steps 4 and 5 until you reach the last coefficient.
7. The final result is the quotient and the remainder.

Using Synthetic Division to Factor the Polynomial

Now that we have learned how to perform synthetic division, let's use it to factor the polynomial $4x^3 + 5x^2 - 18x + 9$. We will consider each of the five possible factors in turn.

Factor 1: $2x - 3$

To use synthetic division to factor the polynomial with the factor $2x - 3$, we need to follow the steps outlined above.

	4	5	-18	9
2	8	14	-6

The final result is $2x^2 + 7x - 3$. This means that the polynomial can be factored as $(2x - 3)(2x^2 + 7x - 3)$.

Factor 2: $x + 3$

To use synthetic division to factor the polynomial with the factor $x + 3$, we need to follow the steps outlined above.

	4	5	-18	9
-3	-12	-10	6

The final result is $4x^2 - 3x + 3$. This means that the polynomial cannot be factored with the factor $x + 3$.

Factor 3: $4x - 3$

To use synthetic division to factor the polynomial with the factor $4x - 3$, we need to follow the steps outlined above.

	4	5	-18	9
-4/3	-16/3	-20/3	8/3

The final result is $4x^2 + 5x - 3$. This means that the polynomial cannot be factored with the factor $4x - 3$.

Factor 4: $x - 1$

To use synthetic division to factor the polynomial with the factor $x - 1$, we need to follow the steps outlined above.

	4	5	-18	9
-1	-4	-5	18

The final result is $4x^2 + 9x - 9$. This means that the polynomial cannot be factored with the factor $x - 1$.

Factor 5: $x - 3$

To use synthetic division to factor the polynomial with the factor $x - 3$, we need to follow the steps outlined above.

	4	5	-18	9
-3	-12	-15	27

The final result is $4x^2 + 5x - 9$. This means that the polynomial cannot be factored with the factor $x - 3$.

Conclusion

In this article, we used synthetic division to find all the factors of the polynomial $4x^3 + 5x^2 - 18x + 9$. We considered five possible factors: $2x - 3$, $x + 3$, $4x - 3$, $x - 1$, and $x - 3$. We found that the polynomial can be factored with the factor $2x - 3$ as $(2x - 3)(2x^2 + 7x - 3)$. We also found that the polynomial cannot be factored with the other four factors.

References

• [1] "Synthetic Division" by Math Is Fun. Retrieved from https://www.mathisfun.com/algebra/synthetic-division.html
• [2] "Factoring Polynomials" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2factors/x2factors-factoring-polynomials

Glossary

• Synthetic Division: A method of dividing polynomials by linear factors.
• Divisor: A linear factor used to divide the polynomial.
• Quotient: The result of dividing the polynomial by the divisor.
• Remainder: The amount left over after dividing the polynomial by the divisor.
  Synthetic Division: A Powerful Tool for Factoring Polynomials - Q&A ===========================================================

Introduction

In our previous article, we used synthetic division to find all the factors of the polynomial $4x^3 + 5x^2 - 18x + 9$. We also discussed the basics of synthetic division and how to perform it. In this article, we will answer some frequently asked questions about synthetic division.

Q&A

Q: What is synthetic division used for?

A: Synthetic division is used to divide polynomials by linear factors. It is a powerful tool for factoring polynomials and is often used in algebra and calculus.

Q: How do I know which factor to use for synthetic division?

A: To determine which factor to use for synthetic division, you need to consider the possible factors of the polynomial. You can use the Rational Root Theorem to find the possible factors.

Q: What is the Rational Root Theorem?

A: The Rational Root Theorem states that if a rational number $p/q$ is a root of the polynomial $a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0$, then $p$ must be a factor of $a_0$ and $q$ must be a factor of $a_n$.

Q: How do I perform synthetic division?

A: 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. Write down the divisor, which is usually a linear factor.
3. Bring down the first coefficient of the polynomial.
4. Multiply the divisor by the first coefficient and write the result below the second coefficient.
5. Add the second coefficient and the result from step 4.
6. Repeat steps 4 and 5 until you reach the last coefficient.
7. The final result is the quotient and the remainder.

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

A: Synthetic division is a shortcut method that eliminates the need for long division. It is faster and more efficient than long division.

Q: Can I use synthetic division to divide polynomials by non-linear factors?

A: No, synthetic division is only used to divide polynomials by linear factors.

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 divisor is a root of the polynomial.

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

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

• Not writing down the coefficients of the polynomial correctly
• Not writing down the divisor correctly
• Not bringing down the first coefficient correctly
• Not multiplying the divisor by the first coefficient correctly
• Not adding the second coefficient and the result from step 4 correctly

Conclusion

In this article, we answered some frequently asked questions about synthetic division. We discussed the basics of synthetic division, how to perform it, and some common mistakes to avoid. We also discussed the Rational Root Theorem and how to use it to find the possible factors of a polynomial.

References

• [1] "Synthetic Division" by Math Is Fun. Retrieved from https://www.mathisfun.com/algebra/synthetic-division.html
• [2] "Factoring Polynomials" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2factors/x2factors-factoring-polynomials

Glossary

• Synthetic Division: A method of dividing polynomials by linear factors.
• Divisor: A linear factor used to divide the polynomial.
• Quotient: The result of dividing the polynomial by the divisor.
• Remainder: The amount left over after dividing the polynomial by the divisor.
• Rational Root Theorem: A theorem that states that if a rational number $p/q$ is a root of the polynomial $a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0$, then $p$ must be a factor of $a_0$ and $q$ must be a factor of $a_n$.