Determine Whether The Second Polynomial Is A Factor Of The First: ${ 5x^2 + 3x + 30 \quad ; \quad X - 2 }$Select The Correct Choice Below And Fill In The Answer Box To Complete Your Choice.A. Yes, Because The Polynomial $[ 5x^2 + 3x +

Introduction

In algebra, determining whether one polynomial is a factor of another is a crucial concept. It involves understanding the relationship between polynomials and their factors. In this article, we will explore how to determine whether the second polynomial is a factor of the first, using the given example: ${ 5x^2 + 3x + 30 \quad ; \quad x - 2 }$. We will delve into the world of polynomial factorization and provide a step-by-step guide on how to determine the correct answer.

Understanding Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as a product of simpler polynomials. This is a fundamental concept in algebra, and it has numerous applications in various fields, including mathematics, science, and engineering. When we factor a polynomial, we are essentially breaking it down into its constituent parts, which can help us solve equations, find roots, and perform other mathematical operations.

The Given Polynomials

In the given example, we have two polynomials:

$${ 5x^2 + 3x + 30 }$$

$${ x - 2 }$$

We need to determine whether the second polynomial, $x - 2$, is a factor of the first polynomial, $5x^2 + 3x + 30$.

The Factor Theorem

The factor theorem states that if a polynomial $f(x)$ is divided by $x - a$, and the remainder is 0, then $x - a$ is a factor of $f(x)$. In other words, if we substitute $a$ into the polynomial and get 0, then $x - a$ is a factor of the polynomial.

Applying the Factor Theorem

To determine whether $x - 2$ is a factor of $5x^2 + 3x + 30$, we need to substitute $x = 2$ into the polynomial and check if the result is 0.

$${ 5(2)^2 + 3(2) + 30 }$$

$${ 5(4) + 6 + 30 }$$

$${ 20 + 6 + 30 }$$

$${ 56 }$$

Since the result is not 0, we cannot conclude that $x - 2$ is a factor of $5x^2 + 3x + 30$.

Using Synthetic Division

Another way to determine whether $x - 2$ is a factor of $5x^2 + 3x + 30$ is to use synthetic division. Synthetic division is a shortcut method for dividing polynomials, and it can help us determine whether a polynomial is a factor of another polynomial.

Here's how to perform synthetic division:

1. Write down the coefficients of the polynomial, in order of decreasing degree.
2. Write down the value of $a$, which is the value we are substituting into the polynomial.
3. Multiply the value of $a$ by the first coefficient, and write down the result.
4. Add the result to the second coefficient, and write down the result.
5. Multiply the value of $a$ by the result, and write down the result.
6. Add the result to the third coefficient, and write down the result.

Using synthetic division, we get:

$${ \begin{array}{c|rrr} 2 & 5 & 3 & 30 \\ & & 10 & 20 \\ \hline & 15 & 50 \\ \end{array} }$$

Since the remainder is not 0, we cannot conclude that $x - 2$ is a factor of $5x^2 + 3x + 30$.

Conclusion

In conclusion, we have explored how to determine whether the second polynomial is a factor of the first, using the given example: ${ 5x^2 + 3x + 30 \quad ; \quad x - 2 }$. We have used the factor theorem and synthetic division to determine whether $x - 2$ is a factor of $5x^2 + 3x + 30$. Since the remainder is not 0 in both cases, we cannot conclude that $x - 2$ is a factor of $5x^2 + 3x + 30$.

Final Answer

The final answer is: No, $x - 2$ is not a factor of $5x^2 + 3x + 30$.

Additional Resources

For more information on polynomial factorization and the factor theorem, please refer to the following resources:

• Khan Academy: Polynomial Factorization
• Mathway: Factor Theorem
• Wolfram Alpha: Polynomial Factorization

References

• "Algebra" by Michael Artin
• "Calculus" by Michael Spivak
• "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
  Polynomial Factorization Q&A =============================

Introduction

In our previous article, we explored how to determine whether one polynomial is a factor of another. We used the factor theorem and synthetic division to determine whether $x - 2$ is a factor of $5x^2 + 3x + 30$. In this article, we will answer some frequently asked questions about polynomial factorization.

Q: What is polynomial factorization?

A: Polynomial factorization is the process of expressing a polynomial as a product of simpler polynomials. This is a fundamental concept in algebra, and it has numerous applications in various fields, including mathematics, science, and engineering.

Q: What is the factor theorem?

A: The factor theorem states that if a polynomial $f(x)$ is divided by $x - a$, and the remainder is 0, then $x - a$ is a factor of $f(x)$. In other words, if we substitute $a$ into the polynomial and get 0, then $x - a$ is a factor of the polynomial.

Q: How do I use synthetic division to factor a polynomial?

A: To use synthetic division to factor a polynomial, follow these steps:

1. Write down the coefficients of the polynomial, in order of decreasing degree.
2. Write down the value of $a$, which is the value we are substituting into the polynomial.
3. Multiply the value of $a$ by the first coefficient, and write down the result.
4. Add the result to the second coefficient, and write down the result.
5. Multiply the value of $a$ by the result, and write down the result.
6. Add the result to the third coefficient, and write down the result.

Q: What is the difference between a factor and a root?

A: A factor is a polynomial that divides another polynomial without leaving a remainder. A root, on the other hand, is a value of $x$ that makes the polynomial equal to 0. For example, if we have the polynomial $x^2 + 4x + 4$, then $x = -2$ is a root of the polynomial, but $x - 2$ is not a factor of the polynomial.

Q: Can a polynomial have multiple factors?

A: Yes, a polynomial can have multiple factors. For example, the polynomial $x^2 + 4x + 4$ can be factored as $(x + 2)^2$. In this case, $x + 2$ is a factor of the polynomial, and it is repeated twice.

Q: How do I determine whether a polynomial is irreducible?

A: A polynomial is irreducible if it cannot be factored into simpler polynomials. To determine whether a polynomial is irreducible, we can use the following methods:

1. Check if the polynomial has any rational roots using the rational root theorem.
2. Use synthetic division to divide the polynomial by possible factors.
3. Check if the polynomial has any repeated factors.

Q: What are some common mistakes to avoid when factoring polynomials?

A: Some common mistakes to avoid when factoring polynomials include:

1. Not checking if the polynomial has any rational roots.
2. Not using synthetic division to divide the polynomial by possible factors.
3. Not checking if the polynomial has any repeated factors.
4. Not simplifying the polynomial after factoring.

Conclusion

In conclusion, we have answered some frequently asked questions about polynomial factorization. We have discussed the factor theorem, synthetic division, and how to determine whether a polynomial is irreducible. We have also highlighted some common mistakes to avoid when factoring polynomials.

Additional Resources

For more information on polynomial factorization, please refer to the following resources:

• Khan Academy: Polynomial Factorization
• Mathway: Factor Theorem
• Wolfram Alpha: Polynomial Factorization

References

• "Algebra" by Michael Artin
• "Calculus" by Michael Spivak
• "Mathematics for Computer Science" by Eric Lehman and Tom Leighton