Which Expression Is The Factorization Of $x^2 + 10x + 21$?A. \[$(x + 3)(x + 7)\$\] B. \[$(x + 4)(x + 6)\$\] C. \[$(x + 6)(x + 15)\$\] D. \[$(x + 7)(x + 14)\$\]
Introduction
Factoring quadratic expressions is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomial expressions. In this article, we will explore the process of factoring quadratic expressions and apply it to the given expression $x^2 + 10x + 21$. We will examine the different options provided and determine which one is the correct factorization of the given expression.
Understanding Quadratic Expressions
A quadratic expression is a polynomial of degree two, which means it has a highest power of two. It can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The expression $x^2 + 10x + 21$ is a quadratic expression, where $a = 1$, $b = 10$, and $c = 21$.
Factoring Quadratic Expressions
Factoring a quadratic expression involves expressing it as a product of two binomial expressions. A binomial expression is a polynomial of degree one, which means it has a highest power of one. The general form of a binomial expression is $ax + b$.
To factor a quadratic expression, we need to find two binomial expressions whose product is equal to the quadratic expression. The process of factoring a quadratic expression involves the following steps:
- Find the factors of the constant term: The constant term is the term that is not multiplied by any variable. In this case, the constant term is $21$. We need to find two numbers whose product is equal to $21$ and whose sum is equal to the coefficient of the linear term, which is $10$.
- Find the factors of the linear term: The linear term is the term that is multiplied by the variable. In this case, the linear term is $10x$. We need to find two numbers whose product is equal to $10$ and whose sum is equal to the coefficient of the linear term, which is $10$.
- Write the factored form: Once we have found the factors of the constant term and the linear term, we can write the factored form of the quadratic expression.
Applying the Factoring Process
Let's apply the factoring process to the given expression $x^2 + 10x + 21$. We need to find two numbers whose product is equal to $21$ and whose sum is equal to $10$. The numbers that satisfy this condition are $3$ and $7$, since $3 \times 7 = 21$ and $3 + 7 = 10$.
Therefore, the factored form of the expression $x^2 + 10x + 21$ is $(x + 3)(x + 7)$.
Evaluating the Options
Now that we have found the correct factorization of the expression $x^2 + 10x + 21$, we can evaluate the options provided.
- Option A: $(x + 3)(x + 7)$
- Option B: $(x + 4)(x + 6)$
- Option C: $(x + 6)(x + 15)$
- Option D: $(x + 7)(x + 14)$
Based on our analysis, we can see that option A is the correct factorization of the expression $x^2 + 10x + 21$.
Conclusion
In this article, we have explored the process of factoring quadratic expressions and applied it to the given expression $x^2 + 10x + 21$. We have found the correct factorization of the expression and evaluated the options provided. The correct factorization of the expression $x^2 + 10x + 21$ is $(x + 3)(x + 7)$.
Final Answer
The final answer is option A: $(x + 3)(x + 7)$.
Additional Tips and Resources
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To factor a quadratic expression, you need to find two binomial expressions whose product is equal to the quadratic expression.
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The process of factoring a quadratic expression involves finding the factors of the constant term and the linear term.
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You can use the factoring process to simplify complex expressions and solve equations.
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For more information on factoring quadratic expressions, you can refer to the following resources:
- Khan Academy: Factoring Quadratic Expressions
- Mathway: Factoring Quadratic Expressions
- Wolfram Alpha: Factoring Quadratic Expressions
Factoring Quadratic Expressions: A Q&A Guide =====================================================
Introduction
In our previous article, we explored the process of factoring quadratic expressions and applied it to the given expression $x^2 + 10x + 21$. We found the correct factorization of the expression and evaluated the options provided. In this article, we will provide a Q&A guide to help you understand the concept of factoring quadratic expressions and apply it to different scenarios.
Q: What is factoring a quadratic expression?
A: Factoring a quadratic expression involves expressing it as a product of two binomial expressions. A binomial expression is a polynomial of degree one, which means it has a highest power of one.
Q: How do I factor a quadratic expression?
A: To factor a quadratic expression, you need to follow these steps:
- Find the factors of the constant term: The constant term is the term that is not multiplied by any variable. You need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.
- Find the factors of the linear term: The linear term is the term that is multiplied by the variable. You need to find two numbers whose product is equal to the linear term and whose sum is equal to the coefficient of the linear term.
- Write the factored form: Once you have found the factors of the constant term and the linear term, you can write the factored form of the quadratic expression.
Q: What are some common mistakes to avoid when factoring quadratic expressions?
A: Some common mistakes to avoid when factoring quadratic expressions include:
- Not finding the correct factors: Make sure you find the correct factors of the constant term and the linear term.
- Not writing the factored form correctly: Make sure you write the factored form of the quadratic expression correctly.
- Not checking the product: Make sure you check the product of the two binomial expressions to ensure it is equal to the original quadratic expression.
Q: How do I check if a quadratic expression is factored correctly?
A: To check if a quadratic expression is factored correctly, you need to follow these steps:
- Multiply the two binomial expressions: Multiply the two binomial expressions to get the original quadratic expression.
- Check if the product is equal to the original quadratic expression: Check if the product of the two binomial expressions is equal to the original quadratic expression.
- Check if the factored form is correct: Check if the factored form of the quadratic expression is correct.
Q: What are some real-world applications of factoring quadratic expressions?
A: Some real-world applications of factoring quadratic expressions include:
- Simplifying complex expressions: Factoring quadratic expressions can help simplify complex expressions and make them easier to work with.
- Solving equations: Factoring quadratic expressions can help solve equations and find the solutions.
- Modeling real-world problems: Factoring quadratic expressions can help model real-world problems and find the solutions.
Q: How do I practice factoring quadratic expressions?
A: To practice factoring quadratic expressions, you can try the following:
- Practice factoring different types of quadratic expressions: Practice factoring different types of quadratic expressions, such as expressions with a leading coefficient of 1 or expressions with a leading coefficient that is not 1.
- Use online resources: Use online resources, such as Khan Academy or Mathway, to practice factoring quadratic expressions.
- Work with a partner or tutor: Work with a partner or tutor to practice factoring quadratic expressions and get feedback on your work.
Conclusion
In this article, we have provided a Q&A guide to help you understand the concept of factoring quadratic expressions and apply it to different scenarios. We have covered common mistakes to avoid, how to check if a quadratic expression is factored correctly, and real-world applications of factoring quadratic expressions. We have also provided tips on how to practice factoring quadratic expressions. By following these tips and practicing factoring quadratic expressions, you can become proficient in factoring quadratic expressions and apply it to different scenarios.