Which Expression Is Equivalent To $9x - (x-8)(x+1) + 10x$?A. $-x^2 + 19x + 8$B. $-x^2 + 26x + 8$C. $8x^2 + 10x + 8$D. $8x^2 + 26x + 8$

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the given expression: $9x - (x-8)(x+1) + 10x$. We will break down the expression into manageable parts, apply the distributive property, and combine like terms to arrive at the simplified expression.

Understanding the Expression

The given expression is a combination of three terms: $9x$, $-(x-8)(x+1)$, and $10x$. To simplify this expression, we need to apply the distributive property to the second term and then combine like terms.

Applying the Distributive Property

The distributive property states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. We can apply this property to the second term, $-(x-8)(x+1)$, by multiplying the two binomials.

$$-(x-8)(x+1) = -(x^2 + x - 8x - 8)$$

Using the distributive property, we can rewrite the expression as:

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

Simplifying further, we get:

$$-x^2 - x + 8x + 8 = -x^2 + 7x + 8$$

Combining Like Terms

Now that we have simplified the second term, we can combine like terms with the first and third terms.

$$9x - (x-8)(x+1) + 10x = 9x - x^2 + 7x + 8 + 10x$$

Combining like terms, we get:

$$9x - x^2 + 7x + 8 + 10x = -x^2 + 26x + 8$$

Conclusion

In this article, we simplified the given algebraic expression by applying the distributive property and combining like terms. We arrived at the simplified expression: $-x^2 + 26x + 8$. This expression is equivalent to the original expression, and it is the correct answer among the given options.

Answer

The correct answer is:

• B. $-x^2 + 26x + 8$

Final Thoughts

Simplifying algebraic expressions is an essential skill for any math enthusiast. By applying the distributive property and combining like terms, we can simplify complex expressions and arrive at the correct answer. In this article, we demonstrated the process of simplifying an algebraic expression, and we hope that this guide will be helpful for anyone struggling with algebraic expressions.

Common Algebraic Identities

Here are some common algebraic identities that you may find useful:

• Distributive Property: $a(b+c) = ab + ac$
• Commutative Property: $a + b = b + a$
• Associative Property: $(a + b) + c = a + (b + c)$
• Identity Property: $a + 0 = a$
• Inverse Property: $a + (-a) = 0$

Practice Problems

Here are some practice problems to help you reinforce your understanding of simplifying algebraic expressions:

• Simplify the expression: $2x - (x-3)(x+2) + 4x$
• Simplify the expression: $3x - (x-2)(x+1) + 5x$
• Simplify the expression: $4x - (x-1)(x+3) + 6x$

Conclusion

Q: What is the distributive property, and how is it used in simplifying algebraic expressions?

A: The distributive property is a fundamental concept in algebra that states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. This property is used to simplify algebraic expressions by multiplying the terms inside the parentheses with the terms outside the parentheses.

Q: How do I apply the distributive property to simplify an algebraic expression?

A: To apply the distributive property, you need to multiply the terms inside the parentheses with the terms outside the parentheses. For example, if you have the expression $-(x-8)(x+1)$, you would multiply the two binomials using the distributive property:

$$-(x-8)(x+1) = -(x^2 + x - 8x - 8)$$

Simplifying further, you get:

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

Q: What is the difference between the distributive property and the commutative property?

A: The distributive property and the commutative property are two different properties in algebra. The distributive property states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. The commutative property, on the other hand, states that for any real numbers $a$ and $b$, $a + b = b + a$.

Q: How do I combine like terms in an algebraic expression?

A: To combine like terms, you need to identify the terms that have the same variable and coefficient. For example, if you have the expression $9x - x^2 + 7x + 8 + 10x$, you can combine the like terms as follows:

$$9x - x^2 + 7x + 8 + 10x = -x^2 + 26x + 8$$

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

A: An algebraic expression is a mathematical statement that contains variables and constants, but it does not have an equal sign. An equation, on the other hand, is a mathematical statement that contains an equal sign and is used to solve for a variable.

Q: How do I simplify an algebraic expression with multiple variables?

A: To simplify an algebraic expression with multiple variables, you need to apply the distributive property and combine like terms. For example, if you have the expression $2x^2y - 3xy^2 + 4x^2y$, you can simplify it as follows:

$$2x^2y - 3xy^2 + 4x^2y = 6x^2y - 3xy^2$$

Q: What are some common algebraic identities that I should know?

A: Some common algebraic identities that you should know include:

• Distributive Property: $a(b+c) = ab + ac$
• Commutative Property: $a + b = b + a$
• Associative Property: $(a + b) + c = a + (b + c)$
• Identity Property: $a + 0 = a$
• Inverse Property: $a + (-a) = 0$

Q: How do I practice simplifying algebraic expressions?

A: To practice simplifying algebraic expressions, you can try the following:

• Start with simple expressions and gradually move on to more complex ones.
• Use online resources, such as algebraic expression simplifiers, to help you practice.
• Work with a partner or join a study group to practice simplifying algebraic expressions together.
• Take online quizzes or tests to assess your understanding of simplifying algebraic expressions.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill for any math enthusiast. By applying the distributive property and combining like terms, we can simplify complex expressions and arrive at the correct answer. We hope that this guide has been helpful in answering your frequently asked questions about simplifying algebraic expressions.