Factor The Trinomial: $\[ A^4 + 71a^2 + 70 \\]

Introduction

Factoring trinomials is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. In this article, we will focus on factoring the trinomial $a^4 + 71a^2 + 70$. This type of trinomial is known as a quadratic trinomial, and it can be factored using various techniques.

Understanding the Trinomial

A trinomial is a polynomial expression with three terms. In the case of the trinomial $a^4 + 71a^2 + 70$, we have three terms: $a^4$, $71a^2$, and $70$. The first two terms are both quadratic expressions, while the third term is a constant.

Factoring Techniques

There are several techniques for factoring trinomials, including:

• Grouping Method: This method involves grouping the first two terms and the last two terms together and then factoring out a common factor.
• Factoring by Difference of Squares: This method involves factoring the trinomial as the difference of two squares.
• Factoring by Grouping and Difference of Squares: This method involves combining the grouping method and the difference of squares method.

Factoring the Trinomial $a^4 + 71a^2 + 70$

To factor the trinomial $a^4 + 71a^2 + 70$, we can use the grouping method. We can group the first two terms and the last two terms together as follows:

$$a^4 + 71a^2 + 70 = (a^4 + 70a^2) + (a^2 + 70)$$

Now, we can factor out a common factor from each group:

$$a^4 + 70a^2 + a^2 + 70 = a^2(a^2 + 70) + 1(a^2 + 70)$$

Next, we can factor out the common factor $(a^2 + 70)$ from each group:

$$a^2(a^2 + 70) + 1(a^2 + 70) = (a^2 + 1)(a^2 + 70)$$

Therefore, the factored form of the trinomial $a^4 + 71a^2 + 70$ is $(a^2 + 1)(a^2 + 70)$.

Conclusion

Factoring trinomials is an essential skill in algebra that involves expressing a quadratic expression as a product of two binomials. In this article, we have focused on factoring the trinomial $a^4 + 71a^2 + 70$ using the grouping method. We have also discussed other factoring techniques, including factoring by difference of squares and factoring by grouping and difference of squares. By mastering these techniques, you can factor trinomials with ease and solve a wide range of algebraic problems.

Common Mistakes to Avoid

When factoring trinomials, there are several common mistakes to avoid:

• Not grouping the terms correctly: Make sure to group the terms in the correct way to factor out a common factor.
• Not factoring out a common factor: Make sure to factor out a common factor from each group to simplify the expression.
• Not checking for factoring by difference of squares: Make sure to check if the trinomial can be factored by difference of squares before using other factoring techniques.

Real-World Applications

Factoring trinomials has numerous real-world applications in various fields, including:

• Science: Factoring trinomials is used to model real-world phenomena, such as the motion of objects under the influence of gravity.
• Engineering: Factoring trinomials is used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Factoring trinomials is used to model economic systems and make predictions about future trends.

Practice Problems

To practice factoring trinomials, try the following problems:

• Factor the trinomial $x^4 + 36x^2 + 35$.
• Factor the trinomial $y^4 + 25y^2 + 24$.
• Factor the trinomial $z^4 + 49z^2 + 48$.

Conclusion

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Q&A: Factoring Trinomials

Q: What is a trinomial?

A: A trinomial is a polynomial expression with three terms.

Q: What are the different types of trinomials?

A: There are two main types of trinomials: quadratic trinomials and cubic trinomials. Quadratic trinomials have the form $ax^2 + bx + c$, while cubic trinomials have the form $ax^3 + bx^2 + cx + d$.

Q: How do I factor a trinomial?

A: There are several techniques for factoring trinomials, including the grouping method, factoring by difference of squares, and factoring by grouping and difference of squares. The choice of technique depends on the specific trinomial being factored.

Q: What is the grouping method?

A: The grouping method involves grouping the first two terms and the last two terms together and then factoring out a common factor.

Q: How do I use the grouping method to factor a trinomial?

A: To use the grouping method, follow these steps:

1. Group the first two terms and the last two terms together.
2. Factor out a common factor from each group.
3. Simplify the expression.

Q: What is factoring by difference of squares?

A: Factoring by difference of squares involves factoring a trinomial as the difference of two squares.

Q: How do I use factoring by difference of squares to factor a trinomial?

A: To use factoring by difference of squares, follow these steps:

1. Check if the trinomial can be factored as the difference of two squares.
2. Factor the trinomial as the difference of two squares.
3. Simplify the expression.

Q: What is factoring by grouping and difference of squares?

A: Factoring by grouping and difference of squares involves combining the grouping method and the difference of squares method.

Q: How do I use factoring by grouping and difference of squares to factor a trinomial?

A: To use factoring by grouping and difference of squares, follow these steps:

1. Group the first two terms and the last two terms together.
2. Check if the trinomial can be factored as the difference of two squares.
3. Factor the trinomial as the difference of two squares.
4. Simplify the expression.

Q: What are some common mistakes to avoid when factoring trinomials?

A: Some common mistakes to avoid when factoring trinomials include:

• Not grouping the terms correctly.
• Not factoring out a common factor.
• Not checking for factoring by difference of squares.

Q: What are some real-world applications of factoring trinomials?

A: Factoring trinomials has numerous real-world applications in various fields, including science, engineering, and economics.

Q: How can I practice factoring trinomials?

A: You can practice factoring trinomials by trying the following problems:

• Factor the trinomial $x^4 + 36x^2 + 35$.
• Factor the trinomial $y^4 + 25y^2 + 24$.
• Factor the trinomial $z^4 + 49z^2 + 48$.

Conclusion

Factoring trinomials is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. In this article, we have provided a comprehensive guide to factoring trinomials, including the grouping method, factoring by difference of squares, and factoring by grouping and difference of squares. We have also discussed common mistakes to avoid and provided real-world applications of factoring trinomials. By mastering these techniques, you can factor trinomials with ease and solve a wide range of algebraic problems.