Simplify The Expression:$\[ \frac{2x^3 - X^2 - 18x + 32}{2x - 6} \\]

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's often required to solve equations and inequalities. In this article, we'll focus on simplifying a given expression by factoring and canceling. We'll break down the process into manageable steps, making it easier to understand and apply.

Understanding the Expression

The given expression is $\frac{2x^3 - x^2 - 18x + 32}{2x - 6}$. To simplify this expression, we need to factor the numerator and denominator, and then cancel out any common factors.

Factoring the Numerator

To factor the numerator, we need to find the greatest common factor (GCF) of the four terms: $2x^3$, $-x^2$, $-18x$, and $32$. The GCF is $2$, so we can factor it out:

$$2(x^3 - \frac{1}{2}x^2 - 9x + 16)$$

Now, we need to factor the quadratic expression inside the parentheses. We can use the quadratic formula or complete the square to find the factors. In this case, we'll use the quadratic formula:

$$x^3 - \frac{1}{2}x^2 - 9x + 16 = (x - 2)(x^2 + 2x - 8)$$

So, the factored form of the numerator is:

$$2(x - 2)(x^2 + 2x - 8)$$

Factoring the Denominator

The denominator is $2x - 6$. We can factor out the GCF, which is $2$, and then factor the remaining expression:

$$2(x - 3)$$

Canceling Common Factors

Now that we have the factored forms of the numerator and denominator, we can cancel out any common factors. In this case, we can cancel out the factor of $2$ and the factor of $(x - 2)$:

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

Simplifying the Expression

We can simplify the expression further by factoring the quadratic expression in the numerator:

$$x^2 + 2x - 8 = (x + 4)(x - 2)$$

So, the simplified form of the expression is:

$$\frac{(x - 2)(x + 4)(x - 2)}{(x - 3)}$$

Canceling Out the Common Factor

We can cancel out the common factor of $(x - 2)$:

$$\frac{(x + 4)(x - 2)}{(x - 3)}$$

Final Answer

The simplified form of the expression is $\frac{(x + 4)(x - 2)}{(x - 3)}$.

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics, and it's often required to solve equations and inequalities. By factoring and canceling, we can simplify complex expressions and make them easier to work with. In this article, we've walked through the process of simplifying a given expression, step by step, and arrived at the final answer.

Tips and Tricks

• Always look for the greatest common factor (GCF) when factoring expressions.
• Use the quadratic formula or complete the square to factor quadratic expressions.
• Cancel out common factors to simplify expressions.
• Check your work by plugging in values or using a calculator.

Common Mistakes to Avoid

• Failing to factor the numerator and denominator.
• Not canceling out common factors.
• Not checking your work.

Real-World Applications

Simplifying algebraic expressions has many real-world applications, including:

• Solving equations and inequalities in physics and engineering.
• Modeling population growth and decay in biology.
• Analyzing data and making predictions in statistics.

Further Reading

For more information on simplifying algebraic expressions, check out the following resources:

• Khan Academy: Simplifying Algebraic Expressions
• Mathway: Simplifying Algebraic Expressions
• Wolfram Alpha: Simplifying Algebraic Expressions

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics, and it's often required to solve equations and inequalities. By following the steps outlined in this article, you can simplify complex expressions and make them easier to work with. Remember to always look for the greatest common factor, use the quadratic formula or complete the square, and cancel out common factors. With practice and patience, you'll become a pro at simplifying algebraic expressions in no time!

Introduction

In our previous article, we walked through the process of simplifying a given expression by factoring and canceling. Now, we'll answer some frequently asked questions (FAQs) about simplifying algebraic expressions.

Q: What is the greatest common factor (GCF) and why is it important?

A: The greatest common factor (GCF) is the largest factor that divides all the terms in an expression. It's essential to find the GCF when factoring expressions because it helps us simplify the expression and cancel out common factors.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you can use the quadratic formula or complete the square. The quadratic formula is:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

where $a$, $b$, and $c$ are the coefficients of the quadratic expression.

Q: What is the difference between factoring and canceling?

A: Factoring involves breaking down an expression into its simplest form by finding the greatest common factor and factoring the remaining expression. Canceling involves removing common factors from the numerator and denominator to simplify the expression.

Q: Can I simplify an expression with a variable in the denominator?

A: Yes, you can simplify an expression with a variable in the denominator. However, you need to be careful when canceling out common factors to avoid dividing by zero.

Q: How do I know when to cancel out common factors?

A: You should cancel out common factors when the expression is in its simplest form and there are no other common factors to cancel out.

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

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

• Failing to factor the numerator and denominator
• Not canceling out common factors
• Not checking your work

Q: How do I check my work when simplifying expressions?

A: You can check your work by plugging in values or using a calculator to evaluate the expression.

Q: What are some real-world applications of simplifying algebraic expressions?

A: Simplifying algebraic expressions has many real-world applications, including:

• Solving equations and inequalities in physics and engineering
• Modeling population growth and decay in biology
• Analyzing data and making predictions in statistics

Q: Where can I find more resources on simplifying algebraic expressions?

A: You can find more resources on simplifying algebraic expressions at:

• Khan Academy: Simplifying Algebraic Expressions
• Mathway: Simplifying Algebraic Expressions
• Wolfram Alpha: Simplifying Algebraic Expressions

Q: Can I simplify an expression with a negative exponent?

A: Yes, you can simplify an expression with a negative exponent. To do this, you need to rewrite the expression with a positive exponent and then simplify.

Q: How do I simplify an expression with a fraction in the numerator or denominator?

A: To simplify an expression with a fraction in the numerator or denominator, you need to find the least common multiple (LCM) of the denominators and rewrite the expression with the LCM as the denominator.

Q: What is the difference between simplifying an expression and solving an equation?

A: Simplifying an expression involves rewriting the expression in its simplest form, while solving an equation involves finding the value of the variable that makes the equation true.

Q: Can I simplify an expression with a variable in the numerator and denominator?

A: Yes, you can simplify an expression with a variable in the numerator and denominator. However, you need to be careful when canceling out common factors to avoid dividing by zero.

Q: How do I know when to use the distributive property when simplifying expressions?

A: You should use the distributive property when simplifying expressions that involve multiplying a binomial by a trinomial or a polynomial.

Q: What are some tips for simplifying expressions with multiple variables?

A: Some tips for simplifying expressions with multiple variables include:

• Use the distributive property to expand the expression
• Factor out common factors
• Cancel out common factors
• Check your work by plugging in values or using a calculator

Q: Can I simplify an expression with a radical in the numerator or denominator?

A: Yes, you can simplify an expression with a radical in the numerator or denominator. To do this, you need to rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator.

Q: How do I simplify an expression with a complex number in the numerator or denominator?

A: To simplify an expression with a complex number in the numerator or denominator, you need to use the conjugate of the complex number to rationalize the denominator.

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

A: Some common mistakes to avoid when simplifying expressions with complex numbers include:

• Failing to rationalize the denominator
• Not using the conjugate of the complex number
• Not checking your work

Q: Where can I find more resources on simplifying expressions with complex numbers?

A: You can find more resources on simplifying expressions with complex numbers at:

• Khan Academy: Simplifying Expressions with Complex Numbers
• Mathway: Simplifying Expressions with Complex Numbers
• Wolfram Alpha: Simplifying Expressions with Complex Numbers