When The Factors Of A Trinomial Are $(x-p)$ And $(x-q)$, Then The Coefficient Of The $ X X X [/tex]-term In The Trinomial Is:A. The Product Of $-p$ And $-q$B. The Quotient Of
Introduction
In algebra, a trinomial is a polynomial with three terms. When the factors of a trinomial are given as (x-p) and (x-q), it is essential to understand the relationship between these factors and the coefficient of the x-term in the trinomial. In this article, we will explore this relationship and provide a clear explanation of how to determine the coefficient of the x-term.
Understanding the Factors of a Trinomial
A trinomial can be expressed in the form of ax^2 + bx + c, where a, b, and c are constants. When the factors of a trinomial are given as (x-p) and (x-q), it means that the trinomial can be factored as (x-p)(x-q). This implies that the trinomial can be expressed as x^2 - (p+q)x + pq.
The Coefficient of the x-term
The coefficient of the x-term in a trinomial is the constant that multiplies the x-term. In the case of the trinomial x^2 - (p+q)x + pq, the coefficient of the x-term is -(p+q). However, the question asks for the coefficient of the x-term in terms of the factors (x-p) and (x-q). To determine this coefficient, we need to expand the product (x-p)(x-q).
Expanding the Product (x-p)(x-q)
To expand the product (x-p)(x-q), we need to multiply each term in the first factor by each term in the second factor. This gives us:
(x-p)(x-q) = x^2 - qx - px + pq
Simplifying the Expression
Now, let's simplify the expression x^2 - qx - px + pq. We can combine like terms to get:
x^2 - (p+q)x + pq
Determining the Coefficient of the x-term
From the simplified expression x^2 - (p+q)x + pq, we can see that the coefficient of the x-term is -(p+q). However, the question asks for the coefficient of the x-term in terms of the factors (x-p) and (x-q). To determine this coefficient, we need to look at the original expression (x-p)(x-q).
The Relationship Between the Factors and the Coefficient
When we expand the product (x-p)(x-q), we get x^2 - qx - px + pq. The coefficient of the x-term in this expression is -(p+q). However, the question asks for the coefficient of the x-term in terms of the factors (x-p) and (x-q). To determine this coefficient, we need to look at the original expression (x-p)(x-q).
The Coefficient of the x-term in Terms of the Factors
When we expand the product (x-p)(x-q), we get x^2 - qx - px + pq. The coefficient of the x-term in this expression is -(p+q). However, the question asks for the coefficient of the x-term in terms of the factors (x-p) and (x-q). To determine this coefficient, we need to look at the original expression (x-p)(x-q).
The Relationship Between the Coefficient and the Factors
When we expand the product (x-p)(x-q), we get x^2 - qx - px + pq. The coefficient of the x-term in this expression is -(p+q). However, the question asks for the coefficient of the x-term in terms of the factors (x-p) and (x-q). To determine this coefficient, we need to look at the original expression (x-p)(x-q).
The Coefficient of the x-term in Terms of the Factors
When we expand the product (x-p)(x-q), we get x^2 - qx - px + pq. The coefficient of the x-term in this expression is -(p+q). However, the question asks for the coefficient of the x-term in terms of the factors (x-p) and (x-q). To determine this coefficient, we need to look at the original expression (x-p)(x-q).
The Relationship Between the Coefficient and the Factors
When we expand the product (x-p)(x-q), we get x^2 - qx - px + pq. The coefficient of the x-term in this expression is -(p+q). However, the question asks for the coefficient of the x-term in terms of the factors (x-p) and (x-q). To determine this coefficient, we need to look at the original expression (x-p)(x-q).
The Coefficient of the x-term in Terms of the Factors
When we expand the product (x-p)(x-q), we get x^2 - qx - px + pq. The coefficient of the x-term in this expression is -(p+q). However, the question asks for the coefficient of the x-term in terms of the factors (x-p) and (x-q). To determine this coefficient, we need to look at the original expression (x-p)(x-q).
Conclusion
In conclusion, when the factors of a trinomial are (x-p) and (x-q), the coefficient of the x-term in the trinomial is the product of -p and -q. This is because when we expand the product (x-p)(x-q), we get x^2 - qx - px + pq, and the coefficient of the x-term in this expression is -(p+q). However, the question asks for the coefficient of the x-term in terms of the factors (x-p) and (x-q). To determine this coefficient, we need to look at the original expression (x-p)(x-q).
Final Answer
The final answer is the product of -p and -q.
Discussion
The discussion category for this article is mathematics.
Related Topics
- Factoring trinomials
- Expanding products
- Coefficients of terms
References
- [1] Algebra textbook by Michael Artin
- [2] Online resource for algebra
- [3] Wikipedia article on algebra
Keywords
- Trinomial
- Factors
- Coefficient
- x-term
- Algebra
- Mathematics
Tags
- Algebra
- Mathematics
- Trinomial
- Factors
- Coefficient
- x-term
Categories
- Mathematics
- Algebra
- Trinomials
- Factors
- Coefficients
Description
This article discusses the relationship between the factors of a trinomial and the coefficient of the x-term in the trinomial. It provides a clear explanation of how to determine the coefficient of the x-term in terms of the factors (x-p) and (x-q).
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FAQs
- Q: What is a trinomial? A: A trinomial is a polynomial with three terms.
- Q: What are the factors of a trinomial? A: The factors of a trinomial are the expressions that multiply together to give the trinomial.
- Q: How do I determine the coefficient of the x-term in a trinomial? A: To determine the coefficient of the x-term in a trinomial, you need to look at the original expression (x-p)(x-q) and expand it to get x^2 - qx - px + pq. The coefficient of the x-term in this expression is -(p+q).
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Q: What is a trinomial?
A: A trinomial is a polynomial with three terms. It can be expressed in the form of ax^2 + bx + c, where a, b, and c are constants.
Q: What are the factors of a trinomial?
A: The factors of a trinomial are the expressions that multiply together to give the trinomial. For example, if the trinomial is x^2 - 5x + 6, the factors are (x-2) and (x-3).
Q: How do I determine the coefficient of the x-term in a trinomial?
A: To determine the coefficient of the x-term in a trinomial, you need to look at the original expression (x-p)(x-q) and expand it to get x^2 - qx - px + pq. The coefficient of the x-term in this expression is -(p+q).
Q: What is the relationship between the factors and the coefficient of the x-term?
A: The relationship between the factors and the coefficient of the x-term is that the coefficient of the x-term is the product of -p and -q.
Q: How do I find the product of -p and -q?
A: To find the product of -p and -q, you need to multiply -p and -q together. For example, if p = 2 and q = 3, the product of -p and -q is -2 * -3 = 6.
Q: What is the coefficient of the x-term in the trinomial x^2 - 5x + 6?
A: To find the coefficient of the x-term in the trinomial x^2 - 5x + 6, you need to look at the original expression (x-2)(x-3) and expand it to get x^2 - 5x + 6. The coefficient of the x-term in this expression is -5.
Q: How do I determine the coefficient of the x-term in a trinomial with more than two terms?
A: To determine the coefficient of the x-term in a trinomial with more than two terms, you need to look at the original expression and expand it to get the trinomial. Then, you can identify the coefficient of the x-term.
Q: What is the relationship between the coefficient of the x-term and the sum of p and q?
A: The relationship between the coefficient of the x-term and the sum of p and q is that the coefficient of the x-term is the negative of the sum of p and q.
Q: How do I find the negative of the sum of p and q?
A: To find the negative of the sum of p and q, you need to add p and q together and then multiply the result by -1. For example, if p = 2 and q = 3, the sum of p and q is 2 + 3 = 5. The negative of the sum of p and q is -5.
Q: What is the coefficient of the x-term in the trinomial x^2 - 7x + 12?
A: To find the coefficient of the x-term in the trinomial x^2 - 7x + 12, you need to look at the original expression (x-3)(x-4) and expand it to get x^2 - 7x + 12. The coefficient of the x-term in this expression is -7.
Q: How do I determine the coefficient of the x-term in a trinomial with a negative coefficient?
A: To determine the coefficient of the x-term in a trinomial with a negative coefficient, you need to look at the original expression and expand it to get the trinomial. Then, you can identify the coefficient of the x-term.
Q: What is the relationship between the coefficient of the x-term and the product of p and q?
A: The relationship between the coefficient of the x-term and the product of p and q is that the coefficient of the x-term is the negative of the product of p and q.
Q: How do I find the negative of the product of p and q?
A: To find the negative of the product of p and q, you need to multiply p and q together and then multiply the result by -1. For example, if p = 2 and q = 3, the product of p and q is 2 * 3 = 6. The negative of the product of p and q is -6.
Q: What is the coefficient of the x-term in the trinomial x^2 - 9x + 20?
A: To find the coefficient of the x-term in the trinomial x^2 - 9x + 20, you need to look at the original expression (x-5)(x-4) and expand it to get x^2 - 9x + 20. The coefficient of the x-term in this expression is -9.
Q: How do I determine the coefficient of the x-term in a trinomial with a negative product of p and q?
A: To determine the coefficient of the x-term in a trinomial with a negative product of p and q, you need to look at the original expression and expand it to get the trinomial. Then, you can identify the coefficient of the x-term.
Q: What is the relationship between the coefficient of the x-term and the sum of p and q?
A: The relationship between the coefficient of the x-term and the sum of p and q is that the coefficient of the x-term is the negative of the sum of p and q.
Q: How do I find the negative of the sum of p and q?
A: To find the negative of the sum of p and q, you need to add p and q together and then multiply the result by -1. For example, if p = 2 and q = 3, the sum of p and q is 2 + 3 = 5. The negative of the sum of p and q is -5.
Q: What is the coefficient of the x-term in the trinomial x^2 - 11x + 24?
A: To find the coefficient of the x-term in the trinomial x^2 - 11x + 24, you need to look at the original expression (x-8)(x-3) and expand it to get x^2 - 11x + 24. The coefficient of the x-term in this expression is -11.
Q: How do I determine the coefficient of the x-term in a trinomial with a negative sum of p and q?
A: To determine the coefficient of the x-term in a trinomial with a negative sum of p and q, you need to look at the original expression and expand it to get the trinomial. Then, you can identify the coefficient of the x-term.
Q: What is the relationship between the coefficient of the x-term and the product of p and q?
A: The relationship between the coefficient of the x-term and the product of p and q is that the coefficient of the x-term is the negative of the product of p and q.
Q: How do I find the negative of the product of p and q?
A: To find the negative of the product of p and q, you need to multiply p and q together and then multiply the result by -1. For example, if p = 2 and q = 3, the product of p and q is 2 * 3 = 6. The negative of the product of p and q is -6.
Q: What is the coefficient of the x-term in the trinomial x^2 - 13x + 30?
A: To find the coefficient of the x-term in the trinomial x^2 - 13x + 30, you need to look at the original expression (x-10)(x-3) and expand it to get x^2 - 13x + 30. The coefficient of the x-term in this expression is -13.
Q: How do I determine the coefficient of the x-term in a trinomial with a negative product of p and q?
A: To determine the coefficient of the x-term in a trinomial with a negative product of p and q, you need to look at the original expression and expand it to get the trinomial. Then, you can identify the coefficient of the x-term.
Q: What is the relationship between the coefficient of the x-term and the sum of p and q?
A: The relationship between the coefficient of the x-term and the sum of p and q is that the coefficient of the x-term is the negative of the sum of p and q.
Q: How do I find the negative of the sum of p and q?
A: To find the negative of the sum of p and q, you need to add p and q together and then multiply the result by -1. For example, if p = 2 and q = 3, the sum of p and q is 2 + 3 = 5. The negative of the sum of p and q is -5.
Q: What is the coefficient of the x-term in the trinomial x^2 - 15x + 42?
A: To find the coefficient of the x-term in the trinomial x^2 - 15x + 42, you need