Factor The Four-term Polynomial By Grouping.${ 7x^2 - 14xy - 3x + 6y }$

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Introduction

In algebra, factoring polynomials is a crucial skill that helps us simplify complex expressions and solve equations. One of the techniques used to factor polynomials is the method of grouping. This method involves grouping terms in a polynomial in a way that allows us to factor out common factors. In this article, we will focus on factoring a four-term polynomial by grouping.

Understanding the Polynomial

The given polynomial is:

7x2−14xy−3x+6y{ 7x^2 - 14xy - 3x + 6y }

This polynomial has four terms, and we need to factor it by grouping. To do this, we need to identify the common factors in the terms.

Step 1: Identify Common Factors

The first step in factoring by grouping is to identify the common factors in the terms. In this case, we can see that the terms have common factors of 7x and 6y.

Step 2: Group the Terms

Now that we have identified the common factors, we can group the terms in a way that allows us to factor out these common factors. We can group the first two terms together and the last two terms together.

(7x2−14xy)+(−3x+6y){ (7x^2 - 14xy) + (-3x + 6y) }

Step 3: Factor Out Common Factors

Now that we have grouped the terms, we can factor out the common factors. In this case, we can factor out 7x from the first group and 6y from the second group.

7x(x−2y)−3(x−2y){ 7x(x - 2y) - 3(x - 2y) }

Step 4: Factor Out the Common Binomial

Now that we have factored out the common factors, we can see that both groups have a common binomial factor of (x - 2y). We can factor this out to get the final factored form of the polynomial.

(7x−3)(x−2y){ (7x - 3)(x - 2y) }

Conclusion

In this article, we have learned how to factor a four-term polynomial by grouping. We identified the common factors in the terms, grouped the terms, factored out the common factors, and finally factored out the common binomial. This technique is useful for simplifying complex expressions and solving equations.

Real-World Applications

Factoring polynomials by grouping has many real-world applications. For example, in physics, we use polynomial equations to model the motion of objects. Factoring these polynomials helps us solve for the unknown variables and understand the behavior of the objects.

Tips and Tricks

Here are some tips and tricks for factoring polynomials by grouping:

  • Identify the common factors in the terms.
  • Group the terms in a way that allows you to factor out the common factors.
  • Factor out the common factors.
  • Factor out the common binomial.

Common Mistakes

Here are some common mistakes to avoid when factoring polynomials by grouping:

  • Not identifying the common factors in the terms.
  • Not grouping the terms correctly.
  • Not factoring out the common factors.
  • Not factoring out the common binomial.

Practice Problems

Here are some practice problems to help you practice factoring polynomials by grouping:

  1. Factor the polynomial: 2x^2 + 5x - 3
  2. Factor the polynomial: x^2 - 4x + 4
  3. Factor the polynomial: 3x^2 - 2x - 5

Solutions

Here are the solutions to the practice problems:

  1. 2x^2 + 5x - 3 = (2x - 1)(x + 3)
  2. x^2 - 4x + 4 = (x - 2)^2
  3. 3x^2 - 2x - 5 = (3x + 1)(x - 5)

Conclusion

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about factoring polynomials by grouping.

Q: What is factoring by grouping?

A: Factoring by grouping is a technique used to factor polynomials by grouping terms in a way that allows us to factor out common factors.

Q: How do I identify the common factors in a polynomial?

A: To identify the common factors in a polynomial, look for terms that have common factors. For example, in the polynomial 7x^2 - 14xy - 3x + 6y, the terms 7x^2 and -14xy have a common factor of 7x.

Q: How do I group the terms in a polynomial?

A: To group the terms in a polynomial, look for terms that have common factors and group them together. For example, in the polynomial 7x^2 - 14xy - 3x + 6y, we can group the first two terms together and the last two terms together.

Q: How do I factor out the common factors?

A: To factor out the common factors, look for the greatest common factor (GCF) of the terms in each group. For example, in the polynomial 7x^2 - 14xy - 3x + 6y, the GCF of the first group is 7x and the GCF of the second group is 6y.

Q: How do I factor out the common binomial?

A: To factor out the common binomial, look for the common binomial factor in both groups. For example, in the polynomial 7x^2 - 14xy - 3x + 6y, the common binomial factor is (x - 2y).

Q: What are some common mistakes to avoid when factoring polynomials by grouping?

A: Some common mistakes to avoid when factoring polynomials by grouping include:

  • Not identifying the common factors in the terms.
  • Not grouping the terms correctly.
  • Not factoring out the common factors.
  • Not factoring out the common binomial.

Q: How can I practice factoring polynomials by grouping?

A: You can practice factoring polynomials by grouping by working through practice problems. Here are some tips to help you practice:

  • Start with simple polynomials and work your way up to more complex ones.
  • Use online resources or worksheets to find practice problems.
  • Work through the problems step by step, identifying the common factors and grouping the terms.
  • Check your work by factoring the polynomial and verifying that it is correct.

Q: What are some real-world applications of factoring polynomials by grouping?

A: Factoring polynomials by grouping has many real-world applications, including:

  • Physics: Factoring polynomials is used to model the motion of objects and solve for unknown variables.
  • Engineering: Factoring polynomials is used to design and optimize systems.
  • Computer Science: Factoring polynomials is used in algorithms and data structures.

Q: Can I use factoring by grouping to factor any polynomial?

A: No, factoring by grouping is not suitable for all polynomials. This technique is best used for polynomials that have common factors and can be grouped in a way that allows us to factor out the common factors.

Q: What are some other techniques for factoring polynomials?

A: Some other techniques for factoring polynomials include:

  • Factoring by greatest common factor (GCF)
  • Factoring by difference of squares
  • Factoring by sum and difference of cubes
  • Factoring by grouping

Conclusion

In conclusion, factoring polynomials by grouping is a useful technique for simplifying complex expressions and solving equations. By identifying the common factors in the terms, grouping the terms, factoring out the common factors, and finally factoring out the common binomial, we can factor polynomials in a way that helps us understand the behavior of the objects. With practice and patience, you can master this technique and apply it to real-world problems.