The Binomial { (y-2)$}$ Is A Factor Of { Y^2 - 10y + 16$}$. What Is The Other Factor?A. { (y-5)$}$ B. { (y+5)$}$ C. { (y-8)$}$ D. { (y+8)$}$

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Introduction

In algebra, factorization is a crucial concept that helps us simplify complex expressions and solve equations. One of the most common types of factorization is the binomial factorization, where we express a quadratic expression as a product of two binomials. In this article, we will explore the binomial factorization problem and learn how to find the other factor when one of the factors is given.

What is Binomial Factorization?

Binomial factorization is a process of expressing a quadratic expression as a product of two binomials. A quadratic expression is a polynomial of degree two, which means it has a highest power of two. For example, the quadratic expression y2−10y+16y^2 - 10y + 16 is a polynomial of degree two. We can express this expression as a product of two binomials, (y−2)(y-2) and (y−8)(y-8), as follows:

y2−10y+16=(y−2)(y−8)y^2 - 10y + 16 = (y-2)(y-8)

The Problem

In this problem, we are given that the binomial (y−2)(y-2) is a factor of the quadratic expression y2−10y+16y^2 - 10y + 16. We need to find the other factor.

Step 1: Understand the Given Factor

The given factor is (y−2)(y-2). This means that the quadratic expression y2−10y+16y^2 - 10y + 16 can be expressed as a product of (y−2)(y-2) and another binomial.

Step 2: Use the Factor Theorem

The factor theorem states that if (y−a)(y-a) is a factor of a polynomial f(y)f(y), then f(a)=0f(a) = 0. In this case, we know that (y−2)(y-2) is a factor of y2−10y+16y^2 - 10y + 16, so we can use the factor theorem to find the other factor.

Step 3: Divide the Quadratic Expression by the Given Factor

To find the other factor, we need to divide the quadratic expression y2−10y+16y^2 - 10y + 16 by the given factor (y−2)(y-2). We can do this using polynomial long division or synthetic division.

Step 4: Perform Polynomial Long Division

Let's perform polynomial long division to divide y2−10y+16y^2 - 10y + 16 by (y−2)(y-2).

  ____________________
y^2 - 10y + 16
-(y^2 - 2y)
  ____________________
  -8y + 16
-( -8y + 16)
  ____________________
  0

Step 5: Identify the Other Factor

From the polynomial long division, we can see that the other factor is (y−8)(y-8).

Conclusion

In this article, we learned how to find the other factor when one of the factors is given. We used the factor theorem and polynomial long division to divide the quadratic expression y2−10y+16y^2 - 10y + 16 by the given factor (y−2)(y-2). The other factor is (y−8)(y-8).

Final Answer

The final answer is:

  • A. {(y-8)$}$

Discussion

This problem is a great example of how to use the factor theorem and polynomial long division to find the other factor when one of the factors is given. The factor theorem is a powerful tool that helps us identify the factors of a polynomial, and polynomial long division is a useful technique for dividing polynomials.

Related Topics

  • Factor Theorem: The factor theorem states that if (y−a)(y-a) is a factor of a polynomial f(y)f(y), then f(a)=0f(a) = 0.
  • Polynomial Long Division: Polynomial long division is a technique for dividing polynomials.
  • Synthetic Division: Synthetic division is a technique for dividing polynomials that is similar to polynomial long division.

References

  • Algebra: Algebra is a branch of mathematics that deals with the study of variables and their relationships.
  • Polynomials: Polynomials are algebraic expressions that consist of variables and coefficients.
  • Factorization: Factorization is the process of expressing a polynomial as a product of simpler polynomials.

FAQs

  • Q: What is the binomial factorization problem? A: The binomial factorization problem is a problem where we need to express a quadratic expression as a product of two binomials.
  • Q: How do we find the other factor when one of the factors is given? A: We can use the factor theorem and polynomial long division to find the other factor.
  • Q: What is the factor theorem? A: The factor theorem states that if (y−a)(y-a) is a factor of a polynomial f(y)f(y), then f(a)=0f(a) = 0.

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Introduction

In our previous article, we explored the binomial factorization problem and learned how to find the other factor when one of the factors is given. In this article, we will continue to provide more information and answer frequently asked questions about the binomial factorization problem.

Q&A

Q: What is the binomial factorization problem?

A: The binomial factorization problem is a problem where we need to express a quadratic expression as a product of two binomials.

Q: How do we find the other factor when one of the factors is given?

A: We can use the factor theorem and polynomial long division to find the other factor.

Q: What is the factor theorem?

A: The factor theorem states that if (y−a)(y-a) is a factor of a polynomial f(y)f(y), then f(a)=0f(a) = 0.

Q: What is polynomial long division?

A: Polynomial long division is a technique for dividing polynomials.

Q: What is synthetic division?

A: Synthetic division is a technique for dividing polynomials that is similar to polynomial long division.

Q: How do we use the factor theorem to find the other factor?

A: We can use the factor theorem to find the other factor by substituting the given factor into the polynomial and setting it equal to zero.

Q: What are some common mistakes to avoid when using the factor theorem?

A: Some common mistakes to avoid when using the factor theorem include:

  • Not substituting the given factor into the polynomial correctly
  • Not setting the polynomial equal to zero correctly
  • Not using the correct values for the variables

Q: How do we use polynomial long division to find the other factor?

A: We can use polynomial long division to find the other factor by dividing the polynomial by the given factor.

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

A: Some common mistakes to avoid when using polynomial long division include:

  • Not dividing the polynomial correctly
  • Not using the correct values for the variables
  • Not simplifying the polynomial correctly

Q: What are some real-world applications of the binomial factorization problem?

A: Some real-world applications of the binomial factorization problem include:

  • Solving quadratic equations
  • Finding the roots of a polynomial
  • Simplifying complex expressions

Conclusion

In this article, we provided more information and answered frequently asked questions about the binomial factorization problem. We hope that this article has been helpful in understanding the binomial factorization problem and how to use the factor theorem and polynomial long division to find the other factor.

Final Answer

The final answer is:

  • A. {(y-8)$}$

Discussion

This problem is a great example of how to use the factor theorem and polynomial long division to find the other factor when one of the factors is given. The factor theorem is a powerful tool that helps us identify the factors of a polynomial, and polynomial long division is a useful technique for dividing polynomials.

Related Topics

  • Factor Theorem: The factor theorem states that if (y−a)(y-a) is a factor of a polynomial f(y)f(y), then f(a)=0f(a) = 0.
  • Polynomial Long Division: Polynomial long division is a technique for dividing polynomials.
  • Synthetic Division: Synthetic division is a technique for dividing polynomials that is similar to polynomial long division.

References

  • Algebra: Algebra is a branch of mathematics that deals with the study of variables and their relationships.
  • Polynomials: Polynomials are algebraic expressions that consist of variables and coefficients.
  • Factorization: Factorization is the process of expressing a polynomial as a product of simpler polynomials.

FAQs

  • Q: What is the binomial factorization problem? A: The binomial factorization problem is a problem where we need to express a quadratic expression as a product of two binomials.
  • Q: How do we find the other factor when one of the factors is given? A: We can use the factor theorem and polynomial long division to find the other factor.
  • Q: What is the factor theorem? A: The factor theorem states that if (y−a)(y-a) is a factor of a polynomial f(y)f(y), then f(a)=0f(a) = 0.

Additional Resources

  • Online Resources: There are many online resources available that provide information and practice problems for the binomial factorization problem.
  • Textbooks: There are many textbooks available that provide information and practice problems for the binomial factorization problem.
  • Tutorials: There are many tutorials available that provide step-by-step instructions for solving the binomial factorization problem.

Conclusion

In conclusion, the binomial factorization problem is a fundamental concept in algebra that helps us express quadratic expressions as a product of two binomials. We hope that this article has been helpful in understanding the binomial factorization problem and how to use the factor theorem and polynomial long division to find the other factor.