The Binomial $(y-2)$ Is A Factor Of $y^2 - 10y + 16$. What Is The Other Factor?A. $ ( Y − 5 ) (y-5) ( Y − 5 ) [/tex] B. $(y+5)$ C. $(y-8)$ D. $ ( Y + 8 ) (y+8) ( Y + 8 ) [/tex]

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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 of quadratic expressions. In this article, we will explore how to factorize a quadratic expression using the binomial factorization method.

What is Binomial Factorization?

Binomial factorization is a method of factorizing a quadratic expression into two binomial factors. A binomial is an algebraic expression consisting of two terms, such as x+yx + y or x−yx - y. The binomial factorization of a quadratic expression is a way of expressing it as a product of two binomials.

The Binomial Factorization Formula

The binomial factorization formula is given by:

ax2+bx+c=(mx+n)(px+q)ax^2 + bx + c = (mx + n)(px + q)

where aa, bb, and cc are constants, and mm, nn, pp, and qq are coefficients.

How to Factorize a Quadratic Expression

To factorize a quadratic expression using the binomial factorization method, we need to follow these steps:

  1. Check if the quadratic expression can be factored: We need to check if the quadratic expression can be factored into two binomials. If it cannot be factored, then we need to use other methods such as completing the square or using the quadratic formula.
  2. Find the factors of the constant term: We need to find the factors of the constant term in the quadratic expression. The constant term is the term that does not have any variable.
  3. Find the factors of the coefficient of the linear term: We need to find the factors of the coefficient of the linear term in the quadratic expression. The linear term is the term that has only one variable.
  4. Combine the factors: We need to combine the factors of the constant term and the coefficient of the linear term to form two binomials.

Example: Factorizing the Quadratic Expression y2−10y+16y^2 - 10y + 16

Let's consider the quadratic expression y2−10y+16y^2 - 10y + 16. We need to factorize this expression using the binomial factorization method.

Step 1: Check if the quadratic expression can be factored

The quadratic expression y2−10y+16y^2 - 10y + 16 can be factored into two binomials.

Step 2: Find the factors of the constant term

The constant term in the quadratic expression is 16. The factors of 16 are 1, 2, 4, 8, and 16.

Step 3: Find the factors of the coefficient of the linear term

The coefficient of the linear term in the quadratic expression is -10. The factors of -10 are 1, -1, 2, -2, 5, and -5.

Step 4: Combine the factors

We need to combine the factors of the constant term and the coefficient of the linear term to form two binomials. We can see that the factors of 16 are 1 and 16, and the factors of -10 are 1 and -10. Therefore, we can write the quadratic expression as:

(y−2)(y−8)(y - 2)(y - 8)

Step 5: Check if the binomial is a factor of the quadratic expression

We are given that the binomial (y−2)(y-2) is a factor of the quadratic expression y2−10y+16y^2 - 10y + 16. We can see that this is true, since we have factored the quadratic expression into two binomials, and one of the binomials is (y−2)(y-2).

Step 6: Find the other factor

We need to find the other factor of the quadratic expression. We can see that the other factor is (y−8)(y-8).

Conclusion

In this article, we have explored how to factorize a quadratic expression using the binomial factorization method. We have also seen how to find the other factor of a quadratic expression, given that one of the binomials is a factor of the quadratic expression. We have used the quadratic expression y2−10y+16y^2 - 10y + 16 as an example to illustrate the steps involved in factorizing a quadratic expression using the binomial factorization method.

Final Answer

The final answer is: (y−8)\boxed{(y-8)}

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Introduction

In our previous article, we explored how to factorize a quadratic expression using the binomial factorization method. We also saw how to find the other factor of a quadratic expression, given that one of the binomials is a factor of the quadratic expression. In this article, we will answer some frequently asked questions about binomial factorization of quadratic expressions.

Q&A

Q: What is the binomial factorization method?

A: The binomial factorization method is a way of expressing a quadratic expression as a product of two binomials.

Q: How do I know if a quadratic expression can be factored using the binomial factorization method?

A: You can check if a quadratic expression can be factored using the binomial factorization method by looking for two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q: What are the steps involved in factorizing a quadratic expression using the binomial factorization method?

A: The steps involved in factorizing a quadratic expression using the binomial factorization method are:

  1. Check if the quadratic expression can be factored.
  2. Find the factors of the constant term.
  3. Find the factors of the coefficient of the linear term.
  4. Combine the factors to form two binomials.

Q: How do I find the other factor of a quadratic expression, given that one of the binomials is a factor of the quadratic expression?

A: To find the other factor of a quadratic expression, given that one of the binomials is a factor of the quadratic expression, you can use the following steps:

  1. Factorize the quadratic expression using the binomial factorization method.
  2. Identify the binomial that is given as a factor of the quadratic expression.
  3. The other factor is the remaining binomial.

Q: What are some common mistakes to avoid when factorizing a quadratic expression using the binomial factorization method?

A: Some common mistakes to avoid when factorizing a quadratic expression using the binomial factorization method are:

  • Not checking if the quadratic expression can be factored.
  • Not finding the factors of the constant term and the coefficient of the linear term.
  • Not combining the factors correctly to form two binomials.

Q: Can the binomial factorization method be used to factorize all quadratic expressions?

A: No, the binomial factorization method cannot be used to factorize all quadratic expressions. The binomial factorization method can only be used to factorize quadratic expressions that can be expressed as a product of two binomials.

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

A: The binomial factorization method has many real-world applications, including:

  • Solving systems of linear equations.
  • Finding the roots of a quadratic equation.
  • Simplifying complex expressions.

Conclusion

In this article, we have answered some frequently asked questions about binomial factorization of quadratic expressions. We have also seen how to factorize a quadratic expression using the binomial factorization method and how to find the other factor of a quadratic expression, given that one of the binomials is a factor of the quadratic expression.

Final Answer

The final answer is: (y−8)\boxed{(y-8)}