Factor The Expression Completely:$3x^2 + 17x + 103x^2$

Introduction

Factoring an expression 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 expression completely, which involves breaking down the expression into its prime factors. We will use the given expression $3x^2 + 17x + 103x^2$ as an example to demonstrate the step-by-step process of factoring.

Understanding the Expression

Before we begin factoring, it's essential to understand the given expression. The expression $3x^2 + 17x + 103x^2$ can be simplified by combining like terms. We can rewrite the expression as:

$$3x^2 + 17x + 103x^2 = (3x^2 + 103x^2) + 17x$$

Using the distributive property, we can simplify the expression further:

$$(3x^2 + 103x^2) + 17x = 106x^2 + 17x$$

Now that we have simplified the expression, we can proceed with factoring.

Factoring the Expression

To factor the expression $106x^2 + 17x$, we need to find two numbers whose product is $106x^2$ and whose sum is $17x$. These numbers are $2x$ and $53x$, since:

$$(2x)(53x) = 106x^2$$

and

$$(2x) + (53x) = 55x$$

However, we need to find two numbers whose sum is $17x$, not $55x$. Therefore, we need to factor out the greatest common factor (GCF) of the expression, which is $17x$.

Factoring Out the GCF

To factor out the GCF, we need to divide each term in the expression by the GCF. In this case, we divide each term by $17x$:

$$\frac{106x^2}{17x} = 6x$$

$$\frac{17x}{17x} = 1$$

Now that we have factored out the GCF, we can rewrite the expression as:

$$106x^2 + 17x = 17x(6x + 1)$$

Factoring the Quadratic Expression

The expression $6x + 1$ is a quadratic expression that cannot be factored further using simple factoring techniques. However, we can use the quadratic formula to find the roots of the expression:

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

In this case, $a = 6$, $b = 1$, and $c = 0$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-1 \pm \sqrt{1^2 - 4(6)(0)}}{2(6)}$$

Simplifying the expression, we get:

$$x = \frac{-1 \pm \sqrt{1}}{12}$$

$$x = \frac{-1 \pm 1}{12}$$

Therefore, the roots of the expression $6x + 1$ are $x = 0$ and $x = -\frac{1}{6}$.

Conclusion

In conclusion, we have factored the expression $3x^2 + 17x + 103x^2$ completely by first simplifying the expression, then factoring out the GCF, and finally factoring the quadratic expression using the quadratic formula. The final factored form of the expression is:

$$3x^2 + 17x + 103x^2 = 17x(6x + 1)$$

Common Mistakes to Avoid

When factoring an expression, it's essential to avoid common mistakes such as:

• Not simplifying the expression before factoring
• Not factoring out the GCF
• Not using the quadratic formula to find the roots of a quadratic expression

By avoiding these common mistakes, you can ensure that you factor expressions correctly and accurately.

Real-World Applications

Factoring expressions has numerous real-world applications in fields such as:

• Physics: Factoring expressions is used to describe the motion of objects under various forces.
• Engineering: Factoring expressions is used to design and optimize systems such as bridges and buildings.
• Economics: Factoring expressions is used to model and analyze economic systems.

In conclusion, factoring expressions is a fundamental concept in algebra that has numerous real-world applications. By understanding the step-by-step process of factoring, you can apply this concept to solve problems in various fields.

Additional Resources

For additional resources on factoring expressions, including video tutorials and practice problems, visit the following websites:

• Khan Academy: Factoring Expressions
• Mathway: Factoring Expressions
• Wolfram Alpha: Factoring Expressions

Introduction

Factoring expressions is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In our previous article, we discussed the step-by-step process of factoring an expression. In this article, we will provide a Q&A guide to help you understand and apply the concepts learned in the previous article.

Q: What is factoring an expression?

A: Factoring an expression involves expressing a polynomial as a product of simpler polynomials. This is done by finding the greatest common factor (GCF) of the expression and factoring it out, or by using the quadratic formula to find the roots of a quadratic expression.

Q: Why is factoring an expression important?

A: Factoring an expression is important because it allows us to simplify complex expressions and solve problems in various fields such as physics, engineering, and economics. It also helps us to identify the roots of a quadratic expression, which is essential in many real-world applications.

Q: How do I factor an expression?

A: To factor an expression, follow these steps:

1. Simplify the expression by combining like terms.
2. Find the greatest common factor (GCF) of the expression.
3. Factor out the GCF.
4. If the expression is a quadratic expression, use the quadratic formula to find the roots.

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

A: The greatest common factor (GCF) of an expression is the largest factor that divides each term in the expression. For example, the GCF of the expression $6x^2 + 12x + 18x^2$ is $6x$, since $6x$ is the largest factor that divides each term.

Q: How do I find the GCF of an expression?

A: To find the GCF of an expression, follow these steps:

1. List the factors of each term in the expression.
2. Identify the common factors among the terms.
3. Choose the largest common factor as the GCF.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula used to find the roots of a quadratic expression. The formula is:

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

Q: How do I use the quadratic formula?

A: To use the quadratic formula, follow these steps:

1. Identify the coefficients of the quadratic expression (a, b, and c).
2. Plug the coefficients into the quadratic formula.
3. Simplify the expression to find the roots.

Q: What are some common mistakes to avoid when factoring an expression?

A: Some common mistakes to avoid when factoring an expression include:

• Not simplifying the expression before factoring
• Not factoring out the GCF
• Not using the quadratic formula to find the roots of a quadratic expression

Q: How do I practice factoring expressions?

A: To practice factoring expressions, try the following:

• Use online resources such as Khan Academy, Mathway, and Wolfram Alpha to practice factoring expressions.
• Work on practice problems to reinforce your understanding of the concepts.
• Apply the concepts learned in this article to real-world problems.

Conclusion

In conclusion, factoring expressions is a fundamental concept in algebra that has numerous real-world applications. By understanding the step-by-step process of factoring and avoiding common mistakes, you can become proficient in factoring expressions and solve problems in various fields. Remember to practice factoring expressions regularly to reinforce your understanding of the concepts.

Additional Resources

For additional resources on factoring expressions, including video tutorials and practice problems, visit the following websites:

• Khan Academy: Factoring Expressions
• Mathway: Factoring Expressions
• Wolfram Alpha: Factoring Expressions

By practicing and applying the concepts learned in this article, you can become proficient in factoring expressions and solve problems in various fields.