Which Of The Binomials Below Is A Factor Of This Trinomial? 5 X 2 − 5 X − 100 5x^2 - 5x - 100 5 X 2 − 5 X − 100 A. X + 5 X + 5 X + 5 B. X − 5 X - 5 X − 5 C. X + 10 X + 10 X + 10 D. X − 4 X - 4 X − 4

Understanding the Problem

To determine which of the given binomials is a factor of the trinomial $5x^2 - 5x - 100$, we need to understand the concept of factoring. Factoring involves expressing a polynomial as a product of simpler polynomials, known as factors. In this case, we are looking for a binomial that, when multiplied by another binomial, results in the given trinomial.

The Importance of Factoring

Factoring is a crucial concept in algebra, and it has numerous applications in various fields, including mathematics, physics, and engineering. It allows us to simplify complex expressions, solve equations, and identify patterns in data. In this article, we will explore the process of factoring and apply it to the given trinomial.

The Given Trinomial

The trinomial we are working with is $5x^2 - 5x - 100$. To factor this expression, we need to find two binomials that, when multiplied together, result in the given trinomial. We will examine each of the given binomials and determine which one is a factor of the trinomial.

Option A: $x + 5$

Let's start by examining the first binomial, $x + 5$. To determine if this binomial is a factor of the trinomial, we need to multiply it by another binomial and see if the result matches the given trinomial.

(x + 5)(Ax + B) = 5x^2 - 5x - 100

Expanding the left-hand side of the equation, we get:

Ax^2 + Bx + 5Ax + 5B = 5x^2 - 5x - 100

Combining like terms, we get:

(A + 5A)x^2 + (B + 5B)x + 5B = 5x^2 - 5x - 100

Simplifying further, we get:

6Ax^2 + (B + 5B)x + 5B = 5x^2 - 5x - 100

Comparing the coefficients of the $x^2$ term, we get:

6A = 5

Solving for A, we get:

A = 5/6

Now, let's compare the coefficients of the $x$ term:

B + 5B = -5

Simplifying, we get:

6B = -5

Solving for B, we get:

B = -5/6

Now that we have found the values of A and B, we can substitute them back into the original equation:

(x + 5)(5/6x - 5/6) = 5x^2 - 5x - 100

Expanding the left-hand side of the equation, we get:

5/6x^2 - 5/6x + 25/6x - 25/6 = 5x^2 - 5x - 100

Combining like terms, we get:

5/6x^2 + 10/6x - 25/6 = 5x^2 - 5x - 100

Multiplying both sides of the equation by 6 to eliminate the fractions, we get:

5x^2 + 10x - 25 = 30x^2 - 30x - 600

Comparing the coefficients of the $x^2$ term, we get:

5 ≠ 30

This means that the binomial $x + 5$ is not a factor of the trinomial $5x^2 - 5x - 100$.

Option B: $x - 5$

Now, let's examine the second binomial, $x - 5$. To determine if this binomial is a factor of the trinomial, we need to multiply it by another binomial and see if the result matches the given trinomial.

(x - 5)(Ax + B) = 5x^2 - 5x - 100

Expanding the left-hand side of the equation, we get:

Ax^2 + Bx - 5Ax - 5B = 5x^2 - 5x - 100

Combining like terms, we get:

(A - 5A)x^2 + (B - 5B)x - 5B = 5x^2 - 5x - 100

Simplifying further, we get:

-4Ax^2 + (B - 5B)x - 5B = 5x^2 - 5x - 100

Comparing the coefficients of the $x^2$ term, we get:

-4A = 5

Solving for A, we get:

A = -5/4

Now, let's compare the coefficients of the $x$ term:

B - 5B = -5

Simplifying, we get:

-4B = -5

Solving for B, we get:

B = 5/4

Now that we have found the values of A and B, we can substitute them back into the original equation:

(x - 5)(-5/4x + 5/4) = 5x^2 - 5x - 100

Expanding the left-hand side of the equation, we get:

-5/4x^2 + 5/4x + 25/4x - 25/4 = 5x^2 - 5x - 100

Combining like terms, we get:

-5/4x^2 + 30/4x - 25/4 = 5x^2 - 5x - 100

Multiplying both sides of the equation by 4 to eliminate the fractions, we get:

-5x^2 + 30x - 25 = 20x^2 - 20x - 400

Comparing the coefficients of the $x^2$ term, we get:

-5 ≠ 20

This means that the binomial $x - 5$ is not a factor of the trinomial $5x^2 - 5x - 100$.

Option C: $x + 10$

Now, let's examine the third binomial, $x + 10$. To determine if this binomial is a factor of the trinomial, we need to multiply it by another binomial and see if the result matches the given trinomial.

(x + 10)(Ax + B) = 5x^2 - 5x - 100

Expanding the left-hand side of the equation, we get:

Ax^2 + Bx + 10Ax + 10B = 5x^2 - 5x - 100

Combining like terms, we get:

(A + 10A)x^2 + (B + 10B)x + 10B = 5x^2 - 5x - 100

Simplifying further, we get:

11Ax^2 + (B + 10B)x + 10B = 5x^2 - 5x - 100

Comparing the coefficients of the $x^2$ term, we get:

11A = 5

Solving for A, we get:

A = 5/11

Now, let's compare the coefficients of the $x$ term:

B + 10B = -5

Simplifying, we get:

11B = -5

Solving for B, we get:

B = -5/11

Now that we have found the values of A and B, we can substitute them back into the original equation:

(x + 10)(5/11x - 5/11) = 5x^2 - 5x - 100

Expanding the left-hand side of the equation, we get:

5/11x^2 - 5/11x + 50/11x - 50/11 = 5x^2 - 5x - 100

Combining like terms, we get:

5/11x^2 + 45/11x - 50/11 = 5x^2 - 5x - 100

Multiplying both sides of the equation by 11 to eliminate the fractions, we get:

5x^2 + 45x - 50 = 55x^2 - 55x - 1100

Comparing the coefficients of the $x^2$ term, we get:

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# **Which of the Binomials Below is a Factor of this Trinomial? - Q&A**
Understanding the Problem
To determine which of the given binomials is a factor of the trinomial $5x^2 - 5x - 100$, we need to understand the concept of factoring. Factoring involves expressing a polynomial as a product of simpler polynomials, known as factors. In this case, we are looking for a binomial that, when multiplied by another binomial, results in the given trinomial.
Q: What is the process of factoring?
A: Factoring involves expressing a polynomial as a product of simpler polynomials, known as factors. To factor a polynomial, we need to find two binomials that, when multiplied together, result in the given polynomial.
Q: How do we determine which binomial is a factor of the trinomial?
A: To determine which binomial is a factor of the trinomial, we need to multiply each binomial by another binomial and see if the result matches the given trinomial.
Q: What are the steps involved in factoring a trinomial?
A: The steps involved in factoring a trinomial are:

Identify the trinomial to be factored.
Determine the possible binomials that could be factors of the trinomial.
Multiply each binomial by another binomial and see if the result matches the given trinomial.
Compare the coefficients of the $x^2$ term, $x$ term, and constant term to determine if the binomial is a factor of the trinomial.

Q: What are the possible binomials that could be factors of the trinomial $5x^2 - 5x - 100$?
A: The possible binomials that could be factors of the trinomial $5x^2 - 5x - 100$ are:

$x + 5$
$x - 5$
$x + 10$
$x - 4$

Q: How do we determine if a binomial is a factor of the trinomial?
A: To determine if a binomial is a factor of the trinomial, we need to multiply the binomial by another binomial and see if the result matches the given trinomial.
Q: What are the advantages of factoring a trinomial?
A: The advantages of factoring a trinomial are:

It allows us to simplify complex expressions.
It helps us to solve equations.
It identifies patterns in data.

Q: What are the common mistakes to avoid when factoring a trinomial?
A: The common mistakes to avoid when factoring a trinomial are:

Not identifying the possible binomials that could be factors of the trinomial.
Not multiplying each binomial by another binomial and comparing the result to the given trinomial.
Not comparing the coefficients of the $x^2$ term, $x$ term, and constant term to determine if the binomial is a factor of the trinomial.

Q: How do we check if a binomial is a factor of the trinomial?
A: To check if a binomial is a factor of the trinomial, we need to multiply the binomial by another binomial and see if the result matches the given trinomial.
Q: What are the possible applications of factoring a trinomial?
A: The possible applications of factoring a trinomial are:

Simplifying complex expressions.
Solving equations.
Identifying patterns in data.

Q: How do we determine if a binomial is a factor of the trinomial $5x^2 - 5x - 100$?
A: To determine if a binomial is a factor of the trinomial $5x^2 - 5x - 100$, we need to multiply each binomial by another binomial and see if the result matches the given trinomial.
Conclusion
In conclusion, factoring a trinomial involves expressing the polynomial as a product of simpler polynomials, known as factors. To determine which binomial is a factor of the trinomial, we need to multiply each binomial by another binomial and see if the result matches the given trinomial. The possible binomials that could be factors of the trinomial $5x^2 - 5x - 100$ are $x + 5$, $x - 5$, $x + 10$, and $x - 4$. We need to compare the coefficients of the $x^2$ term, $x$ term, and constant term to determine if the binomial is a factor of the trinomial.