Simplify The Expression:${ \frac{3x^2 - 29x + 18}{3x - 2} }$

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the techniques involved in factoring and canceling to simplify complex expressions. In this article, we will focus on simplifying the given expression: $\frac{3x^2 - 29x + 18}{3x - 2}$. We will break down the steps involved in factoring the numerator and denominator, and then cancel out any common factors to simplify the expression.

Factoring the Numerator

To simplify the given expression, we first need to factor the numerator, which is $3x^2 - 29x + 18$. Factoring a quadratic expression involves finding two numbers whose product is the constant term (in this case, 18) and whose sum is the coefficient of the linear term (in this case, -29). We can start by listing the factors of 18 and finding the pair that adds up to -29.

Factors of 18

The factors of 18 are: 1, 2, 3, 6, 9, 18

Finding the Correct Pair

We need to find a pair of factors that add up to -29. Let's try different combinations:

• 1 + 18 = 19 (not equal to -29)
• 2 + 16 = 18 (not equal to -29)
• 3 + 15 = 18 (not equal to -29)
• 6 + 12 = 18 (not equal to -29)
• 9 + 9 = 18 (not equal to -29)
• 18 + 1 = 19 (not equal to -29)

However, we can also try to factor the numerator by grouping. We can rewrite the numerator as:

$3x^2 - 29x + 18 = (3x^2 - 24x) - (5x - 18)$

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

$(3x^2 - 24x) - (5x - 18) = 3x(x - 8) - 5(x - 8)$

We can now factor out the common term $(x - 8)$:

$3x(x - 8) - 5(x - 8) = (3x - 5)(x - 8)$

So, the factored form of the numerator is $(3x - 5)(x - 8)$.

Factoring the Denominator

The denominator is $3x - 2$. We can factor out a common term of 1:

$3x - 2 = 3(x - \frac{2}{3})$

However, we can also rewrite the denominator as:

$3x - 2 = 3x - 2$

Canceling Common Factors

Now that we have factored the numerator and denominator, we can cancel out any common factors. In this case, we have:

$\frac{(3x - 5)(x - 8)}{3(x - \frac{2}{3})}$

We can cancel out the common factor of $(x - 8)$:

$\frac{(3x - 5)(x - 8)}{3(x - \frac{2}{3})} = \frac{3x - 5}{3(x - \frac{2}{3})}$

We can also cancel out the common factor of 3:

$\frac{3x - 5}{3(x - \frac{2}{3})} = \frac{x - \frac{5}{3}}{x - \frac{2}{3}}$

Simplifying the Expression

Now that we have canceled out the common factors, we can simplify the expression:

$\frac{x - \frac{5}{3}}{x - \frac{2}{3}} = \frac{x - \frac{5}{3}}{x - \frac{2}{3}} \cdot \frac{3}{3}$

We can now simplify the expression by multiplying the numerator and denominator by 3:

$\frac{x - \frac{5}{3}}{x - \frac{2}{3}} \cdot \frac{3}{3} = \frac{3x - 5}{3x - 2}$

Conclusion

In this article, we simplified the given expression $\frac{3x^2 - 29x + 18}{3x - 2}$ by factoring the numerator and denominator, and then canceling out any common factors. We used the techniques of factoring and canceling to simplify the expression and arrive at the final answer. This article demonstrates the importance of understanding the techniques involved in simplifying algebraic expressions, and how they can be applied to solve complex problems in mathematics.

Final Answer

The final answer is $\boxed{\frac{3x - 5}{3x - 2}}$.

Introduction

In our previous article, we simplified the expression $\frac{3x^2 - 29x + 18}{3x - 2}$ by factoring the numerator and denominator, and then canceling out any common factors. In this article, we will provide a Q&A guide to help you understand the techniques involved in simplifying algebraic expressions.

Q: What is factoring in algebra?

A: Factoring in algebra involves breaking down an expression into simpler components, such as the product of two or more factors. For example, the expression $6x^2 + 12x$ can be factored as $6x(x + 2)$.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term. For example, the expression $x^2 + 5x + 6$ can be factored as $(x + 3)(x + 2)$.

Q: What is canceling in algebra?

A: Canceling in algebra involves eliminating common factors between the numerator and denominator of a fraction. For example, the expression $\frac{2x}{2x}$ can be canceled to $\frac{x}{x}$, which simplifies to $1$.

Q: How do I cancel common factors in a fraction?

A: To cancel common factors in a fraction, you need to identify the common factors between the numerator and denominator and eliminate them. For example, the expression $\frac{6x}{2x}$ can be canceled to $\frac{3}{1}$, which simplifies to $3$.

Q: What is the difference between factoring and canceling?

A: Factoring involves breaking down an expression into simpler components, while canceling involves eliminating common factors between the numerator and denominator of a fraction.

Q: How do I know when to factor and when to cancel?

A: You should factor when you have a complex expression that can be broken down into simpler components, and you should cancel when you have a fraction with common factors between the numerator and denominator.

Q: Can I simplify an expression by canceling without factoring?

A: No, you cannot simplify an expression by canceling without factoring. Canceling requires that you have a fraction with common factors between the numerator and denominator.

Q: Can I factor an expression without canceling?

A: Yes, you can factor an expression without canceling. Factoring involves breaking down an expression into simpler components, regardless of whether there are common factors between the numerator and denominator.

Q: How do I know if an expression can be simplified by canceling?

A: You can check if an expression can be simplified by canceling by looking for common factors between the numerator and denominator.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Not factoring the numerator and denominator
• Not canceling common factors
• Canceling factors that are not common to both the numerator and denominator
• Not simplifying the expression after canceling

Conclusion

In this article, we provided a Q&A guide to help you understand the techniques involved in simplifying algebraic expressions. We covered topics such as factoring, canceling, and common mistakes to avoid. By following these guidelines, you can simplify complex expressions and arrive at the final answer.

Final Answer

The final answer is $\boxed{\frac{3x - 5}{3x - 2}}$.