What Is The Completely Factored Form Of The Expression $16x^2 + 8x + 32?$A. $8(2x^2 + X + 4$\]B. $4(12x^2 + 4x + 28$\]C. $8x(8x^2 + X + 24$\]

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Introduction

In mathematics, factoring an expression is a process of expressing it as a product of simpler expressions, called factors. Factoring is an essential skill in algebra, as it allows us to simplify complex expressions, solve equations, and understand the underlying structure of mathematical relationships. In this article, we will explore the completely factored form of the expression $16x^2 + 8x + 32$.

Understanding the Expression

The given expression is a quadratic expression in the form of $ax^2 + bx + c$. In this case, $a = 16$, $b = 8$, and $c = 32$. To factor this expression, we need to find two binomials whose product is equal to the given expression.

Factoring the Expression

To factor the expression $16x^2 + 8x + 32$, we can start by looking for common factors. We can see that all three terms have a common factor of $8$. Factoring out $8$ from each term, we get:

$$16x^2 + 8x + 32 = 8(2x^2 + x + 4)$$

However, this is not the completely factored form of the expression. We need to factor the quadratic expression inside the parentheses.

Factoring the Quadratic Expression

The quadratic expression $2x^2 + x + 4$ can be factored using various methods, such as factoring by grouping or using the quadratic formula. However, in this case, we can try to factor it by looking for two binomials whose product is equal to the quadratic expression.

After some trial and error, we can see that the quadratic expression can be factored as:

$$2x^2 + x + 4 = (2x + 4)(x + 1)$$

Completely Factored Form

Now that we have factored the quadratic expression, we can write the completely factored form of the original expression:

$$16x^2 + 8x + 32 = 8(2x + 4)(x + 1)$$

Conclusion

In this article, we have explored the completely factored form of the expression $16x^2 + 8x + 32$. We started by factoring out a common factor of $8$ from each term, and then factored the quadratic expression inside the parentheses. The completely factored form of the expression is $8(2x + 4)(x + 1)$.

Comparison with Answer Choices

Now that we have found the completely factored form of the expression, we can compare it with the answer choices provided:

A. $8(2x^2 + x + 4)$ B. $4(12x^2 + 4x + 28)$ C. $8x(8x^2 + x + 24)$

We can see that answer choice A is close to the completely factored form we found, but it is not exactly the same. Answer choice B is incorrect, as it has a different coefficient and a different quadratic expression. Answer choice C is also incorrect, as it has a different coefficient and a different quadratic expression.

Final Answer

Based on our analysis, the completely factored form of the expression $16x^2 + 8x + 32$ is:

$$8(2x + 4)(x + 1)$$

This is the correct answer.

Frequently Asked Questions

• What is the completely factored form of the expression $16x^2 + 8x + 32$? The completely factored form of the expression $16x^2 + 8x + 32$ is $8(2x + 4)(x + 1)$.
• How do I factor a quadratic expression? There are various methods to factor a quadratic expression, such as factoring by grouping or using the quadratic formula.
• What is the difference between factoring and simplifying an expression? Factoring an expression involves expressing it as a product of simpler expressions, while simplifying an expression involves combining like terms to reduce its complexity.

Related Topics

• Factoring quadratic expressions
• Simplifying expressions
• Algebraic manipulations

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
  

Introduction

Factoring quadratic expressions is an essential skill in algebra, and it can be a challenging task for many students. In this article, we will answer some of the most frequently asked questions about factoring quadratic expressions.

Q&A

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

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

Q: How do I factor a quadratic expression?

A: There are various methods to factor a quadratic expression, such as factoring by grouping or using the quadratic formula. You can also try to factor the expression by looking for two binomials whose product is equal to the quadratic expression.

Q: What is the completely factored form of the expression $16x^2 + 8x + 32$?

A: The completely factored form of the expression $16x^2 + 8x + 32$ is $8(2x + 4)(x + 1)$.

Q: How do I determine if a quadratic expression can be factored?

A: You can try to factor the quadratic expression by looking for two binomials whose product is equal to the quadratic expression. If you cannot find two binomials, you can try to use the quadratic formula to find the roots of the quadratic expression.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that allows you to find the roots of a quadratic expression. The quadratic formula is given by:

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

Q: How do I use the quadratic formula to find the roots of a quadratic expression?

A: To use the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the quadratic expression. Then, you can plug these values into the quadratic formula to find the roots of the quadratic expression.

Q: What is the difference between a quadratic expression and a quadratic equation?

A: A quadratic expression is an algebraic expression that contains a quadratic term, while a quadratic equation is an equation that contains a quadratic term. A quadratic equation is typically written in the form $ax^2 + bx + c = 0$.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can try to factor the quadratic expression, or you can use the quadratic formula to find the roots of the quadratic expression.

Q: What is the significance of factoring quadratic expressions?

A: Factoring quadratic expressions is an essential skill in algebra, and it can be used to solve quadratic equations, simplify expressions, and understand the underlying structure of mathematical relationships.

Conclusion

In this article, we have answered some of the most frequently asked questions about factoring quadratic expressions. We have discussed the difference between factoring and simplifying an expression, the various methods to factor a quadratic expression, and the significance of factoring quadratic expressions.

Related Topics

• Factoring quadratic expressions
• Simplifying expressions
• Algebraic manipulations
• Quadratic equations
• Quadratic formula

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Additional Resources

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