Factor This Polynomial Expression: 2 ( 3 X − 2 ) 2 + 9 ( 3 X − 2 ) − 5 2(3x-2)^2 + 9(3x-2) - 5 2 ( 3 X − 2 ) 2 + 9 ( 3 X − 2 ) − 5

Introduction

Polynomial expressions are a fundamental concept in algebra, and factoring them is a crucial skill to master. In this article, we will explore how to factor the polynomial expression $2(3x-2)^2 + 9(3x-2) - 5$. We will break down the process into manageable steps, making it easy to understand and follow along.

Understanding the Expression

Before we dive into factoring, let's take a closer look at the given expression:

$2(3x-2)^2 + 9(3x-2) - 5$

This expression consists of three terms:

1. $2(3x-2)^2$
2. $9(3x-2)$
3. $-5$

Step 1: Identify the Common Factor

The first step in factoring is to identify any common factors among the terms. In this case, we can see that each term contains the factor $(3x-2)$. We can rewrite the expression as:

$2(3x-2)^2 + 9(3x-2) - 5 = (3x-2)(2(3x-2) + 9) - 5$

Step 2: Factor the Quadratic Expression

Now that we have identified the common factor, we can focus on factoring the quadratic expression $2(3x-2) + 9$. This expression can be factored as:

$2(3x-2) + 9 = 6x - 4 + 9 = 6x + 5$

So, the expression becomes:

$(3x-2)(6x + 5) - 5$

Step 3: Factor the Difference of Squares

The expression $(3x-2)(6x + 5)$ can be factored using the difference of squares formula:

$a^2 - b^2 = (a + b)(a - b)$

In this case, we have:

$(3x-2)(6x + 5) = (3x-2 + 6x + 5)(3x-2 - 6x - 5)$

Simplifying the expression, we get:

$(3x-2 + 6x + 5)(3x-2 - 6x - 5) = (9x + 3)(-3x - 7)$

So, the expression becomes:

$(9x + 3)(-3x - 7) - 5$

Step 4: Simplify the Expression

The final step is to simplify the expression by combining like terms:

$(9x + 3)(-3x - 7) - 5 = -27x^2 - 63x - 21x - 21 - 5$

Combining like terms, we get:

$-27x^2 - 84x - 26$

Conclusion

Factoring the polynomial expression $2(3x-2)^2 + 9(3x-2) - 5$ requires a step-by-step approach. By identifying the common factor, factoring the quadratic expression, and using the difference of squares formula, we were able to simplify the expression to $-27x^2 - 84x - 26$. This process demonstrates the importance of breaking down complex expressions into manageable parts and using algebraic techniques to simplify them.

Tips and Tricks

• When factoring polynomial expressions, always look for common factors among the terms.
• Use the difference of squares formula to factor expressions of the form $a^2 - b^2$.
• Simplify expressions by combining like terms.

Practice Problems

• Factor the polynomial expression $3x^2 + 12x + 15$.
• Factor the polynomial expression $2x^2 - 5x - 3$.

References

• Algebraic Expressions
• Factoring Polynomials

Glossary

• Polynomial Expression: An expression consisting of variables and coefficients combined using addition, subtraction, and multiplication.
• Factoring: The process of expressing a polynomial expression as a product of simpler expressions.
• Difference of Squares: A formula used to factor expressions of the form $a^2 - b^2$.
  Factor this Polynomial Expression: A Q&A Guide =====================================================

Introduction

In our previous article, we explored how to factor the polynomial expression $2(3x-2)^2 + 9(3x-2) - 5$. We broke down the process into manageable steps, making it easy to understand and follow along. In this article, we will answer some of the most frequently asked questions about factoring polynomial expressions.

Q&A

Q: What is factoring in algebra?

A: Factoring is the process of expressing a polynomial expression as a product of simpler expressions. It involves breaking down a complex expression into its constituent parts, making it easier to solve and manipulate.

Q: How do I know if a polynomial expression can be factored?

A: To determine if a polynomial expression can be factored, look for common factors among the terms. If you can identify a common factor, you can factor the expression using that factor.

Q: What are some common factoring techniques?

A: Some common factoring techniques include:

• Greatest Common Factor (GCF): This involves finding the largest factor that divides all the terms in the expression.
• Difference of Squares: This involves factoring expressions of the form $a^2 - b^2$ using the formula $(a + b)(a - b)$.
• Quadratic Formula: This involves factoring quadratic expressions using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, look for two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term. These numbers are the roots of the quadratic equation.

Q: What is the difference between factoring and simplifying?

A: Factoring involves expressing a polynomial expression as a product of simpler expressions, while simplifying involves combining like terms to reduce the complexity of the expression.

Q: Can I factor a polynomial expression with a negative sign?

A: Yes, you can factor a polynomial expression with a negative sign. Simply treat the negative sign as a factor and factor the expression as usual.

Q: How do I factor a polynomial expression with a variable in the denominator?

A: To factor a polynomial expression with a variable in the denominator, first simplify the expression by combining like terms. Then, factor the expression as usual.

Q: Can I factor a polynomial expression with a fraction?

A: Yes, you can factor a polynomial expression with a fraction. Simply treat the fraction as a single term and factor the expression as usual.

Practice Problems

• Factor the polynomial expression $x^2 + 5x + 6$.
• Factor the polynomial expression $2x^2 - 3x - 1$.
• Factor the polynomial expression $x^2 + 2x - 3$.

Tips and Tricks

• Always look for common factors among the terms when factoring a polynomial expression.
• Use the difference of squares formula to factor expressions of the form $a^2 - b^2$.
• Simplify expressions by combining like terms.

References

• Algebraic Expressions
• Factoring Polynomials

Glossary

• Polynomial Expression: An expression consisting of variables and coefficients combined using addition, subtraction, and multiplication.
• Factoring: The process of expressing a polynomial expression as a product of simpler expressions.
• Difference of Squares: A formula used to factor expressions of the form $a^2 - b^2$.
• Greatest Common Factor (GCF): The largest factor that divides all the terms in a polynomial expression.
• Quadratic Formula: A formula used to factor quadratic expressions.