What Is The Factorization Of The Trinomial Below?$x^2 - 6x + 5$A. $(x-5)(x-1$\]B. $(x+4)(x-2$\]C. $(x+5)(x-1$\]D. $(x-4)(x-2$\]

Feb 28, 2025 by ADMIN 128 views

Understanding Trinomials and Factorization

A trinomial is a polynomial expression consisting of three terms. Factorization of a trinomial involves expressing it as a product of two binomials. This process is essential in algebra, as it helps in solving equations, finding roots, and simplifying expressions. In this article, we will focus on factorizing the given trinomial, $x^2 - 6x + 5$ , and determine the correct factorization among the provided options.

The Factorization Process

To factorize a trinomial, we need to find two binomials whose product equals the given trinomial. The general form of a trinomial is $ax^2 + bx + c$ . We can factorize it using the following steps:

Find two numbers whose product is $ac$ and whose sum is $b$ : In this case, $a = 1$ , $b = -6$ , and $c = 5$ . We need to find two numbers whose product is $1 \times 5 = 5$ and whose sum is $-6$ .
Write the two binomials: Once we have found the two numbers, we can write the two binomials in the form $(x + p)(x + q)$ , where $p$ and $q$ are the numbers we found in step 1.

Applying the Factorization Process

Let's apply the factorization process to the given trinomial, $x^2 - 6x + 5$ . We need to find two numbers whose product is $1 \times 5 = 5$ and whose sum is $-6$ . The two numbers are $-5$ and $-1$ , as their product is $5$ and their sum is $-6$ .

Writing the Two Binomials

Now that we have found the two numbers, $-5$ and $-1$ , we can write the two binomials in the form $(x + p)(x + q)$ . In this case, the two binomials are $(x - 5)(x - 1)$ .

Evaluating the Options

We have determined that the correct factorization of the trinomial $x^2 - 6x + 5$ is $(x - 5)(x - 1)$ . Let's evaluate the options provided:

Option A: $(x-5)(x-1)$
Option B: $(x+4)(x-2)$
Option C: $(x+5)(x-1)$
Option D: $(x-4)(x-2)$

Conclusion

Based on our analysis, the correct factorization of the trinomial $x^2 - 6x + 5$ is Option A: $(x-5)(x-1)$ . This is because the product of the two binomials equals the given trinomial, and the sum of the coefficients of the two binomials is equal to the coefficient of the middle term of the trinomial.

Final Answer

The final answer is Option A: $(x-5)(x-1)$ .

Understanding Trinomial Factorization

Trinomial factorization is a process of expressing a trinomial as a product of two binomials. This process is essential in algebra, as it helps in solving equations, finding roots, and simplifying expressions. In this article, we will address some frequently asked questions (FAQs) on trinomial factorization.

Q: What is a trinomial?

A: A trinomial is a polynomial expression consisting of three terms. It can be written in the form $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I factorize a trinomial?

A: To factorize a trinomial, you need to find two binomials whose product equals the given trinomial. The general form of a trinomial is $ax^2 + bx + c$ . You can factorize it using the following steps:

Find two numbers whose product is $ac$ and whose sum is $b$ : In this case, $a = 1$ , $b = -6$ , and $c = 5$ . You need to find two numbers whose product is $1 \times 5 = 5$ and whose sum is $-6$ .
Write the two binomials: Once you have found the two numbers, you can write the two binomials in the form $(x + p)(x + q)$ , where $p$ and $q$ are the numbers you found in step 1.

Q: What are the common mistakes to avoid while factorizing a trinomial?

A: Some common mistakes to avoid while factorizing a trinomial include:

Not finding the correct numbers: Make sure to find the correct numbers whose product is $ac$ and whose sum is $b$ .
Not writing the binomials correctly: Make sure to write the binomials in the correct form, $(x + p)(x + q)$ .
Not checking the product: Make sure to check the product of the two binomials to ensure it equals the given trinomial.

Q: How do I determine the correct factorization of a trinomial?

A: To determine the correct factorization of a trinomial, you need to check the product of the two binomials to ensure it equals the given trinomial. You can also use the following steps:

Check the product: Multiply the two binomials to ensure the product equals the given trinomial.
Check the sum: Check the sum of the coefficients of the two binomials to ensure it equals the coefficient of the middle term of the trinomial.

Q: What are some examples of trinomial factorization?

A: Some examples of trinomial factorization include:

$x^2 - 6x + 5$ : This trinomial can be factorized as $(x - 5)(x - 1)$ .
$x^2 + 8x + 15$ : This trinomial can be factorized as $(x + 3)(x + 5)$ .
$x^2 - 9x + 20$ : This trinomial can be factorized as $(x - 4)(x - 5)$ .

Q: How do I use trinomial factorization in real-life situations?

A: Trinomial factorization is used in various real-life situations, including:

Solving equations: Trinomial factorization can be used to solve equations involving trinomials.
Finding roots: Trinomial factorization can be used to find the roots of a trinomial.
Simplifying expressions: Trinomial factorization can be used to simplify expressions involving trinomials.

Conclusion

Trinomial factorization is a powerful tool in algebra that helps in solving equations, finding roots, and simplifying expressions. By understanding the process of trinomial factorization and avoiding common mistakes, you can determine the correct factorization of a trinomial and use it in various real-life situations.