Which Of The Binomials Below Is A Factor Of This Trinomial?${5x^2 + 20x + 15}$A. { X-1$}$ B. { X-5$}$ C. { X+5$}$ D. { X+1$}$

Introduction

In algebra, factoring is a process of expressing a polynomial as a product of simpler polynomials. This is a crucial concept in mathematics, as it allows us to solve equations and inequalities more easily. In this article, we will explore which of the given binomials is a factor of the trinomial $5x^2 + 20x + 15$.

Understanding the Trinomial

The trinomial $5x^2 + 20x + 15$ is a quadratic expression that can be factored into the product of two binomials. To factor this trinomial, we need to find two numbers whose product is $5 \times 15 = 75$ and whose sum is $20$. These numbers are $25$ and $3$, as $25 \times 3 = 75$ and $25 + 3 = 28$, which is close to $20$ but not exactly. However, we can see that $25$ and $3$ are not the correct numbers, but we can use them to find the correct numbers.

Let's try to factor the trinomial by grouping. We can write the trinomial as $(5x^2 + 25x) + (20x - 25x) + 15$. Now, we can factor out the common terms: $5x(x + 5) + 5x(4 - 5) + 15$. This simplifies to $5x(x + 5) - 5x + 15$.

Now, we can see that the trinomial can be factored as $5x(x + 5) - 5(x - 3)$. This means that the trinomial can be written as $(5x - 5)(x + 3)$.

Analyzing the Binomials

Now that we have factored the trinomial, we can analyze the given binomials to see which one is a factor of the trinomial.

• Option A: $x - 1$. This binomial is not a factor of the trinomial, as it does not divide the trinomial evenly.
• Option B: $x - 5$. This binomial is not a factor of the trinomial, as it does not divide the trinomial evenly.
• Option C: $x + 5$. This binomial is a factor of the trinomial, as it divides the trinomial evenly.
• Option D: $x + 1$. This binomial is not a factor of the trinomial, as it does not divide the trinomial evenly.

Conclusion

In conclusion, the binomial $x + 5$ is a factor of the trinomial $5x^2 + 20x + 15$. This is because the trinomial can be written as $(5x - 5)(x + 3)$, and $x + 5$ is one of the factors.

Final Answer

Introduction

In our previous article, we explored which of the given binomials is a factor of the trinomial $5x^2 + 20x + 15$. We found that the binomial $x + 5$ is a factor of the trinomial. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is factoring in algebra?

A: Factoring is a process of expressing a polynomial as a product of simpler polynomials. This is a crucial concept in mathematics, as it allows us to solve equations and inequalities more easily.

Q: How do I factor a trinomial?

A: To factor a trinomial, you need to find two numbers whose product is the product of the first and last terms, and whose sum is the middle term. You can also use the method of grouping to factor a trinomial.

Q: What is the method of grouping?

A: The method of grouping is a technique used to factor a trinomial. It involves writing the trinomial as the sum of two binomials, and then factoring out the common terms.

Q: How do I know which binomial is a factor of the trinomial?

A: To determine which binomial is a factor of the trinomial, you need to check if the binomial divides the trinomial evenly. If it does, then it is a factor of the trinomial.

Q: What is the difference between a factor and a multiple?

A: A factor is a polynomial that divides another polynomial evenly, while a multiple is a polynomial that is the product of another polynomial and an integer.

Q: Can a trinomial have more than one factor?

A: Yes, a trinomial can have more than one factor. In fact, a trinomial can have multiple factors, depending on the specific trinomial.

Q: How do I find all the factors of a trinomial?

A: To find all the factors of a trinomial, you need to use the method of grouping and factor out the common terms. You can also use the factoring formulas to find the factors of a trinomial.

Q: What are the factoring formulas?

A: The factoring formulas are:

• $a^2 + 2ab + b^2 = (a + b)^2$
• $a^2 - 2ab + b^2 = (a - b)^2$
• $a^2 + b^2 = (a + bi)(a - bi)$

Q: Can I use the factoring formulas to factor a trinomial?

A: Yes, you can use the factoring formulas to factor a trinomial. However, you need to make sure that the trinomial can be written in the form of the formula.

Q: What is the difference between factoring and simplifying?

A: Factoring involves expressing a polynomial as a product of simpler polynomials, while simplifying involves combining like terms to reduce the complexity of a polynomial.

Q: Can I simplify a trinomial before factoring it?

A: Yes, you can simplify a trinomial before factoring it. In fact, simplifying a trinomial can make it easier to factor.

Q: How do I simplify a trinomial?

A: To simplify a trinomial, you need to combine like terms. This involves adding or subtracting the coefficients of the like terms.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, $2x$ and $5x$ are like terms, while $2x$ and $3y$ are not like terms.

Q: Can I simplify a trinomial with variables?

A: Yes, you can simplify a trinomial with variables. In fact, simplifying a trinomial with variables can make it easier to factor.

Q: How do I simplify a trinomial with variables?

A: To simplify a trinomial with variables, you need to combine like terms. This involves adding or subtracting the coefficients of the like terms.

Q: What is the final answer?

A: The final answer is C.