Factor The Polynomial: $2x(x-3)+9(x-3)$A. $(x-3)(2x+9)$ B. $ 18 X ( X − 3 ) 18x(x-3) 18 X ( X − 3 ) [/tex] C. $(2x+3)(x-9)$ D. $(x-3)(2x-9)$

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Introduction

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we will focus on factoring the given polynomial: $2x(x-3)+9(x-3)$. We will explore the different methods of factoring and provide a step-by-step guide on how to factor the polynomial.

What is Factoring?

Factoring is the process of expressing a polynomial as a product of simpler polynomials. It involves finding the factors of the polynomial, which are the numbers or expressions that multiply together to give the original polynomial. Factoring is an essential concept in algebra, as it allows us to simplify complex expressions and solve equations.

Types of Factoring

There are several types of factoring, including:

  • Greatest Common Factor (GCF) Factoring: This involves finding the greatest common factor of the terms in the polynomial and factoring it out.
  • Difference of Squares Factoring: This involves factoring the difference of two squares, which is a polynomial of the form a2−b2a^2 - b^2.
  • Sum and Difference Factoring: This involves factoring the sum or difference of two terms, which is a polynomial of the form a+ba + b or a−ba - b.
  • Quadratic Factoring: This involves factoring a quadratic polynomial, which is a polynomial of the form ax2+bx+cax^2 + bx + c.

Factoring the Polynomial

Now that we have discussed the different types of factoring, let's focus on factoring the given polynomial: $2x(x-3)+9(x-3)$. To factor this polynomial, we can use the GCF factoring method.

Step 1: Identify the GCF

The first step in factoring the polynomial is to identify the greatest common factor (GCF) of the terms. In this case, the GCF is (x−3)(x-3).

Step 2: Factor out the GCF

Once we have identified the GCF, we can factor it out of each term. This involves dividing each term by the GCF and multiplying the result by the GCF.

from sympy import symbols, factor

x = symbols('x')


poly = 2x(x-3) + 9*(x-3)



factored_poly = factor(poly)


print(factored_poly)

Step 3: Simplify the Expression

Once we have factored out the GCF, we can simplify the expression by combining like terms.

from sympy import symbols, factor, simplify

x = symbols('x')



poly = 2x(x-3) + 9*(x-3)



factored_poly = factor(poly)



simplified_poly = simplify(factored_poly)


print(simplified_poly)

Conclusion

In this article, we have discussed the concept of factoring polynomials and provided a step-by-step guide on how to factor the given polynomial: $2x(x-3)+9(x-3)$. We have used the GCF factoring method to factor the polynomial and simplified the expression by combining like terms. Factoring polynomials is an essential concept in algebra, as it allows us to simplify complex expressions and solve equations.

Answer

The correct answer is:

A. $(x-3)(2x+9)$

Discussion

This problem requires the student to factor the given polynomial using the GCF factoring method. The student must identify the GCF, factor it out of each term, and simplify the expression by combining like terms. This problem requires the student to have a strong understanding of factoring polynomials and to be able to apply the GCF factoring method.

Tips and Tricks

  • Make sure to identify the GCF correctly before factoring it out of each term.
  • Factor out the GCF from each term by dividing each term by the GCF and multiplying the result by the GCF.
  • Simplify the expression by combining like terms.

Practice Problems

  1. Factor the polynomial: $3x(x+2)+4(x+2)$
  2. Factor the polynomial: $2x(x-5)+3(x-5)$
  3. Factor the polynomial: $x(x+1)+2(x+1)$

Conclusion

Introduction

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In our previous article, we discussed the concept of factoring polynomials and provided a step-by-step guide on how to factor the given polynomial: $2x(x-3)+9(x-3)$. In this article, we will provide a Q&A guide to help you understand the concept of factoring polynomials and to answer some common questions related to factoring.

Q: What is factoring?

A: Factoring is the process of expressing a polynomial as a product of simpler polynomials. It involves finding the factors of the polynomial, which are the numbers or expressions that multiply together to give the original polynomial.

Q: Why is factoring important?

A: Factoring is an essential concept in algebra because it allows us to simplify complex expressions and solve equations. By factoring a polynomial, we can break it down into simpler components, making it easier to work with and solve.

Q: What are the different types of factoring?

A: There are several types of factoring, including:

  • Greatest Common Factor (GCF) Factoring: This involves finding the greatest common factor of the terms in the polynomial and factoring it out.
  • Difference of Squares Factoring: This involves factoring the difference of two squares, which is a polynomial of the form a2−b2a^2 - b^2.
  • Sum and Difference Factoring: This involves factoring the sum or difference of two terms, which is a polynomial of the form a+ba + b or a−ba - b.
  • Quadratic Factoring: This involves factoring a quadratic polynomial, which is a polynomial of the form ax2+bx+cax^2 + bx + c.

Q: How do I factor a polynomial?

A: To factor a polynomial, you need to follow these steps:

  1. Identify the greatest common factor (GCF) of the terms.
  2. Factor out the GCF from each term.
  3. Simplify the expression by combining like terms.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest number or expression that divides each term in the polynomial without leaving a remainder.

Q: How do I find the GCF?

A: To find the GCF, you need to list the factors of each term and find the largest common factor.

Q: What is the difference of squares?

A: The difference of squares is a polynomial of the form a2−b2a^2 - b^2, which can be factored as (a+b)(a−b)(a+b)(a-b).

Q: How do I factor a difference of squares?

A: To factor a difference of squares, you need to follow these steps:

  1. Identify the difference of squares.
  2. Factor it as (a+b)(a−b)(a+b)(a-b).

Q: What is the sum and difference?

A: The sum and difference are polynomials of the form a+ba + b or a−ba - b, which can be factored as (a+b)(a+b) or (a−b)(a-b).

Q: How do I factor a sum and difference?

A: To factor a sum and difference, you need to follow these steps:

  1. Identify the sum or difference.
  2. Factor it as (a+b)(a+b) or (a−b)(a-b).

Q: What is quadratic factoring?

A: Quadratic factoring is the process of factoring a quadratic polynomial, which is a polynomial of the form ax2+bx+cax^2 + bx + c.

Q: How do I factor a quadratic polynomial?

A: To factor a quadratic polynomial, you need to follow these steps:

  1. Identify the quadratic polynomial.
  2. Factor it as (ax+b)(cx+d)(ax+b)(cx+d).

Conclusion

Factoring polynomials is an essential concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we have provided a Q&A guide to help you understand the concept of factoring polynomials and to answer some common questions related to factoring. We hope that this guide has been helpful in clarifying the concept of factoring polynomials.

Practice Problems

  1. Factor the polynomial: $3x(x+2)+4(x+2)$
  2. Factor the polynomial: $2x(x-5)+3(x-5)$
  3. Factor the polynomial: $x(x+1)+2(x+1)$

Tips and Tricks

  • Make sure to identify the greatest common factor (GCF) correctly before factoring it out of each term.
  • Factor out the GCF from each term by dividing each term by the GCF and multiplying the result by the GCF.
  • Simplify the expression by combining like terms.

Common Mistakes

  • Failing to identify the greatest common factor (GCF) correctly.
  • Factoring out the wrong term.
  • Failing to simplify the expression by combining like terms.

Conclusion

Factoring polynomials is an essential concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we have provided a Q&A guide to help you understand the concept of factoring polynomials and to answer some common questions related to factoring. We hope that this guide has been helpful in clarifying the concept of factoring polynomials.