The Expression $y^2 - 10y + 24$ Has A Factor Of $y - 4$. What Is Another Factor Of $y^2 - 10y + 24$?A. \$y - 20$[/tex\] B. $y - 14$ C. $y - 8$ D. \$y - 6$[/tex\]
Introduction
In algebra, factorization is a crucial concept that helps us simplify complex expressions and solve equations. Given a quadratic expression, we can often factor it into simpler expressions, making it easier to work with. In this article, we will explore the factorization of the expression $y^2 - 10y + 24$, which has a known factor of $y - 4$. Our goal is to find another factor of this expression.
Understanding the Given Factor
The given factor is $y - 4$. This means that when we substitute $y = 4$ into the original expression, the result should be zero. Let's verify this:
As expected, the expression equals zero when $y = 4$. This confirms that $y - 4$ is indeed a factor of the expression.
Using Polynomial Division
To find another factor, we can use polynomial division. We will divide the original expression by the given factor, $y - 4$. This process involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend.
Let's perform the polynomial division:
The result of the division is $y - 6$. This means that $y - 6$ is another factor of the expression $y^2 - 10y + 24$.
Conclusion
In this article, we used the given factor $y - 4$ to find another factor of the expression $y^2 - 10y + 24$. By performing polynomial division, we discovered that $y - 6$ is indeed another factor of the expression. This factorization helps us simplify the expression and makes it easier to work with.
Answer
The correct answer is:
- C. $y - 8$ is not the correct answer, as we found that the correct factor is $y - 6$.
Final Thoughts
Introduction
In our previous article, we explored the factorization of the expression $y^2 - 10y + 24$, which has a known factor of $y - 4$. We used polynomial division to find another factor of the expression, which turned out to be $y - 6$. In this article, we will address some common questions and provide additional insights into the factorization process.
Q&A
Q: What is the significance of the given factor in the factorization process?
A: The given factor, in this case, $y - 4$, is a crucial component in the factorization process. It helps us identify the other factor by performing polynomial division. The given factor is essentially a "seed" that allows us to uncover the hidden factor.
Q: How do I know when to stop the polynomial division process?
A: You can stop the polynomial division process when the remainder is zero or when the degree of the remainder is less than the degree of the divisor. In our case, the remainder was zero, which confirmed that $y - 6$ is indeed another factor of the expression.
Q: Can I use other methods to find the other factor, such as factoring by grouping or using the quadratic formula?
A: Yes, you can use other methods to find the other factor, but polynomial division is often the most straightforward approach. Factoring by grouping or using the quadratic formula may be more complex and may not always yield the correct result.
Q: What if the given factor is not a linear expression, but rather a quadratic expression?
A: If the given factor is a quadratic expression, you can use polynomial division to find the other factor. However, you may need to use more advanced techniques, such as synthetic division or long division, to perform the division.
Q: Can I use the factorization process to find the roots of the expression?
A: Yes, the factorization process can help you find the roots of the expression. By setting each factor equal to zero, you can solve for the roots of the expression.
Q: What if I'm given a cubic or higher-degree expression to factorize?
A: If you're given a cubic or higher-degree expression to factorize, you may need to use more advanced techniques, such as synthetic division or the rational root theorem, to find the factors.
Conclusion
In this article, we addressed some common questions and provided additional insights into the factorization process. By understanding the significance of the given factor and using polynomial division, we can uncover hidden factors and make the expression more manageable. Whether you're working with linear or quadratic expressions, the factorization process is a powerful tool that can help you simplify complex expressions and solve equations.
Final Thoughts
Factorization is a fundamental concept in algebra that has numerous applications in mathematics and other fields. By mastering the factorization process, you can develop a deeper understanding of mathematical concepts and improve your problem-solving skills. Whether you're a student or a professional, the factorization process is an essential tool that can help you tackle complex mathematical problems with confidence.
Additional Resources
Answer Key
- Q1: The given factor is a crucial component in the factorization process.
- Q2: You can stop the polynomial division process when the remainder is zero or when the degree of the remainder is less than the degree of the divisor.
- Q3: Yes, you can use other methods to find the other factor, but polynomial division is often the most straightforward approach.
- Q4: If the given factor is a quadratic expression, you can use polynomial division to find the other factor.
- Q5: Yes, the factorization process can help you find the roots of the expression.
- Q6: If you're given a cubic or higher-degree expression to factorize, you may need to use more advanced techniques, such as synthetic division or the rational root theorem.