Factor The Trinomial Into The Product Of Two Binomials: $\[ X^2 - 2x + 1 \\]A. \[$(x-1)(x-1)\$\]B. \[$(x-2)(x+1)\$\]C. \[$(x+2)(x-1)\$\]D. \[$(x-2)(x-2)\$\]
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Understanding Trinomials and Binomials
In algebra, a trinomial is a polynomial with three terms, while a binomial is a polynomial with two terms. Factoring a trinomial into the product of two binomials is an essential skill in algebra, as it allows us to simplify complex expressions and solve equations. In this article, we will explore how to factor a trinomial into the product of two binomials.
The General Form of a Trinomial
A trinomial can be written in the general form:
ax^2 + bx + c
where a, b, and c are constants, and x is the variable. To factor a trinomial, we need to find two binomials whose product equals the original trinomial.
The Formula for Factoring a Trinomial
The formula for factoring a trinomial is:
ax^2 + bx + c = (mx + n)(px + q)
where m, n, p, and q are constants. To factor a trinomial, we need to find the values of m, n, p, and q that satisfy the equation.
Example: Factoring the Trinomial x^2 - 2x + 1
Let's consider the trinomial x^2 - 2x + 1. We can write it in the general form as:
x^2 - 2x + 1 = ax^2 + bx + c
where a = 1, b = -2, and c = 1. To factor this trinomial, we need to find two binomials whose product equals the original trinomial.
Step 1: Find the Factors of the Constant Term
The constant term is 1, which has only one factor, 1. So, we can write:
x^2 - 2x + 1 = (x + ?)(x + ?)
Step 2: Find the Factors of the Coefficient of the x Term
The coefficient of the x term is -2, which has two factors, -1 and 2. So, we can write:
x^2 - 2x + 1 = (x - 1)(x - 1)
Step 3: Check the Answer
We can check our answer by multiplying the two binomials:
(x - 1)(x - 1) = x^2 - 2x + 1
This confirms that our answer is correct.
Conclusion
Factoring a trinomial into the product of two binomials requires us to find two binomials whose product equals the original trinomial. We can use the formula ax^2 + bx + c = (mx + n)(px + q) to factor a trinomial. By following the steps outlined in this article, we can factor a trinomial into the product of two binomials.
Answer Key
The correct answer is:
A. (x - 1)(x - 1)
This is the only option that matches the factored form of the trinomial x^2 - 2x + 1.
Tips and Tricks
- To factor a trinomial, we need to find two binomials whose product equals the original trinomial.
- We can use the formula ax^2 + bx + c = (mx + n)(px + q) to factor a trinomial.
- To find the factors of the constant term, we can list all the factors of the constant term.
- To find the factors of the coefficient of the x term, we can list all the factors of the coefficient of the x term.
- We can check our answer by multiplying the two binomials.
Practice Problems
- Factor the trinomial x^2 + 5x + 6.
- Factor the trinomial x^2 - 7x + 12.
- Factor the trinomial x^2 + 2x - 15.
Solutions
- x^2 + 5x + 6 = (x + 2)(x + 3)
- x^2 - 7x + 12 = (x - 3)(x - 4)
- x^2 + 2x - 15 = (x + 5)(x - 3)
Conclusion
Factoring a trinomial into the product of two binomials is an essential skill in algebra. By following the steps outlined in this article, we can factor a trinomial into the product of two binomials. We can use the formula ax^2 + bx + c = (mx + n)(px + q) to factor a trinomial. By practicing the examples and solutions provided in this article, we can become proficient in factoring trinomials.
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Frequently Asked Questions
Q: What is a trinomial?
A: A trinomial is a polynomial with three terms. It can be written in the general form as ax^2 + bx + c, where a, b, and c are constants, and x is the variable.
Q: How do I factor a trinomial into the product of two binomials?
A: To factor a trinomial into the product of two binomials, we need to find two binomials whose product equals the original trinomial. We can use the formula ax^2 + bx + c = (mx + n)(px + q) to factor a trinomial.
Q: What is the formula for factoring a trinomial?
A: The formula for factoring a trinomial is:
ax^2 + bx + c = (mx + n)(px + q)
where m, n, p, and q are constants.
Q: How do I find the factors of the constant term?
A: To find the factors of the constant term, we can list all the factors of the constant term. For example, if the constant term is 6, we can list the factors as 1, 2, 3, and 6.
Q: How do I find the factors of the coefficient of the x term?
A: To find the factors of the coefficient of the x term, we can list all the factors of the coefficient of the x term. For example, if the coefficient of the x term is 5, we can list the factors as 1 and 5.
Q: How do I check my answer?
A: To check your answer, you can multiply the two binomials and see if the product equals the original trinomial.
Q: What are some common mistakes to avoid when factoring trinomials?
A: Some common mistakes to avoid when factoring trinomials include:
- Not using the correct formula
- Not finding the correct factors of the constant term
- Not finding the correct factors of the coefficient of the x term
- Not checking the answer
Q: How can I practice factoring trinomials?
A: You can practice factoring trinomials by working through examples and solutions, such as those provided in this article. You can also try factoring trinomials on your own and checking your answers.
Q: What are some real-world applications of factoring trinomials?
A: Factoring trinomials has many real-world applications, including:
- Simplifying complex expressions
- Solving equations
- Graphing functions
- Analyzing data
Conclusion
Factoring trinomials into the product of two binomials is an essential skill in algebra. By following the steps outlined in this article, we can factor a trinomial into the product of two binomials. We can use the formula ax^2 + bx + c = (mx + n)(px + q) to factor a trinomial. By practicing the examples and solutions provided in this article, we can become proficient in factoring trinomials.
Additional Resources
- Khan Academy: Factoring Trinomials
- Mathway: Factoring Trinomials
- Wolfram Alpha: Factoring Trinomials
Practice Problems
- Factor the trinomial x^2 + 5x + 6.
- Factor the trinomial x^2 - 7x + 12.
- Factor the trinomial x^2 + 2x - 15.
Solutions
- x^2 + 5x + 6 = (x + 2)(x + 3)
- x^2 - 7x + 12 = (x - 3)(x - 4)
- x^2 + 2x - 15 = (x + 5)(x - 3)
Conclusion
Factoring trinomials into the product of two binomials is an essential skill in algebra. By following the steps outlined in this article, we can factor a trinomial into the product of two binomials. We can use the formula ax^2 + bx + c = (mx + n)(px + q) to factor a trinomial. By practicing the examples and solutions provided in this article, we can become proficient in factoring trinomials.