Grace, Chelsea, And Roan Are Simplifying The Same Polynomial Expression. Which Student's Work Is Correct And Why?Given Expression: ${3(2-x)-2(6x-8)}$Student Work:Grace's Work1. ${3(2-x)-2(6x-8)}$2. ${=6-3x-12x+16}$3.

Introduction

Simplifying polynomial expressions is a fundamental concept in algebra that requires careful attention to detail and a thorough understanding of the order of operations. In this article, we will examine the work of three students, Grace, Chelsea, and Roan, as they simplify the same polynomial expression. Our goal is to determine whose work is correct and why.

The Given Expression

The given expression is:

$$3(2-x)-2(6x-8)$$

This expression involves the multiplication of a constant by a binomial and the subtraction of a product of a constant and a binomial.

Student Work: Grace

Grace's work is as follows:

1. $$3(2-x)-2(6x-8)$$

2. $$=6-3x-12x+16$$

3. $$=6-15x+16$$

4. $$=22-15x$$

Student Work: Chelsea

Chelsea's work is as follows:

1. $$3(2-x)-2(6x-8)$$

2. $$=6-3x-12x+16$$

3. $$=6-15x+16$$

4. $$=22-15x$$

Student Work: Roan

Roan's work is as follows:

1. $$3(2-x)-2(6x-8)$$

2. $$=6-3x-12x+16$$

3. $$=6-15x+16$$

4. $$=22-15x$$

Analysis

At first glance, it appears that all three students have arrived at the same solution. However, upon closer inspection, we can see that there is a subtle difference in their work.

Step 1: Distributive Property

The distributive property states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. In this case, we can apply the distributive property to the first term, $3(2-x)$, to get:

$$3(2-x) = 3(2) - 3(x) = 6 - 3x$$

Similarly, we can apply the distributive property to the second term, $-2(6x-8)$, to get:

$$-2(6x-8) = -2(6x) + 2(8) = -12x + 16$$

Step 2: Combining Like Terms

Now that we have applied the distributive property, we can combine like terms. In this case, we have:

$$6 - 3x - 12x + 16$$

We can combine the like terms, $-3x$ and $-12x$, to get:

$$6 - 15x + 16$$

Conclusion

Based on our analysis, we can see that all three students have made the same mistake. They have not applied the distributive property correctly, and as a result, their work is incorrect.

The correct solution is:

$$3(2-x)-2(6x-8)$$

$$=6-3x-12x+16$$

$$=6-15x+16$$

$$=22-15x$$

However, this is not the final answer. We can simplify the expression further by combining the like terms, $6$ and $16$, to get:

$$=22-15x$$

$$=22-15x$$

This is the final answer.

Why is Roan's Work Incorrect?

Roan's work is incorrect because he has not applied the distributive property correctly. In the first step, he has not distributed the $3$ to the terms inside the parentheses, and as a result, his work is incorrect.

Why is Chelsea's Work Incorrect?

Chelsea's work is incorrect because she has not applied the distributive property correctly. In the first step, she has not distributed the $3$ to the terms inside the parentheses, and as a result, her work is incorrect.

Why is Grace's Work Incorrect?

Grace's work is incorrect because she has not applied the distributive property correctly. In the first step, she has not distributed the $3$ to the terms inside the parentheses, and as a result, her work is incorrect.

Conclusion

In conclusion, none of the students' work is correct. They have all made the same mistake, which is not applying the distributive property correctly. The correct solution is:

$$3(2-x)-2(6x-8)$$

$$=6-3x-12x+16$$

$$=6-15x+16$$

$$=22-15x$$

This is the final answer.

Final Answer

$$

Q: What is the distributive property?

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 means that we can distribute a single term to multiple terms inside parentheses.

Q: How do I apply the distributive property?

A: To apply the distributive property, simply multiply the single term to each term inside the parentheses. For example, if we have $3(2-x)$, we can apply the distributive property to get $3(2) - 3(x) = 6 - 3x$.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when simplifying an expression. The order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify a polynomial expression?

A: To simplify a polynomial expression, follow these steps:

1. Apply the distributive property to any terms that are multiplied by a single term.
2. Combine like terms.
3. Simplify any exponential expressions.
4. Evaluate any multiplication and division operations from left to right.
5. Finally, evaluate any addition and subtraction operations from left to right.

Q: What are like terms?

A: Like terms are terms that have the same variable(s) raised to the same power. For example, $2x$ and $5x$ are like terms because they both have the variable $x$ raised to the power of 1.

Q: How do I combine like terms?

A: To combine like terms, simply add or subtract the coefficients of the like terms. For example, if we have $2x + 5x$, we can combine the like terms to get $7x$.

Q: What is the final answer to the given expression?

A: The final answer to the given expression is $\boxed{22-15x}$.

Q: Why is it important to simplify polynomial expressions?

A: Simplifying polynomial expressions is important because it allows us to:

• Evaluate expressions more easily
• Solve equations more easily
• Understand the behavior of functions more easily
• Make predictions about real-world phenomena more easily

Q: Can you provide more examples of simplifying polynomial expressions?

A: Yes, here are a few more examples:

• Simplify the expression $2(3x-4) + 5(2x+3)$.
• Simplify the expression $3(2x^2-5x+1) - 2(3x^2+4x-2)$.
• Simplify the expression $4x^2 + 3x - 2 + 2x^2 - 5x + 1$.

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

A: You should use the distributive property whenever you have a single term multiplied by a binomial or a trinomial. For example, if you have $3(2x-4)$, you should use the distributive property to get $6x - 12$.

Q: Can you provide more information about the order of operations?

A: Yes, here are some additional resources that provide more information about the order of operations:

• Khan Academy: Order of Operations
• Mathway: Order of Operations
• Purplemath: Order of Operations

Q: How do I evaluate expressions with exponents?

A: To evaluate expressions with exponents, follow these steps:

1. Evaluate any exponential expressions.
2. Simplify any terms that have exponents.
3. Evaluate any multiplication and division operations from left to right.
4. Finally, evaluate any addition and subtraction operations from left to right.

Q: Can you provide more examples of evaluating expressions with exponents?

A: Yes, here are a few more examples:

• Evaluate the expression $2^3 + 3^2$.
• Evaluate the expression $4^2 - 2^3$.
• Evaluate the expression $3^2 + 2^3$.

Q: How do I simplify expressions with fractions?

A: To simplify expressions with fractions, follow these steps:

1. Simplify any terms that have fractions.
2. Combine like terms.
3. Simplify any exponential expressions.
4. Evaluate any multiplication and division operations from left to right.
5. Finally, evaluate any addition and subtraction operations from left to right.

Q: Can you provide more examples of simplifying expressions with fractions?

A: Yes, here are a few more examples:

• Simplify the expression $\frac{1}{2} + \frac{1}{4}$.
• Simplify the expression $\frac{3}{4} - \frac{1}{2}$.
• Simplify the expression $\frac{2}{3} + \frac{1}{6}$.

Q: How do I evaluate expressions with absolute value?

A: To evaluate expressions with absolute value, follow these steps:

1. Evaluate any terms that have absolute value.
2. Simplify any terms that have absolute value.
3. Evaluate any multiplication and division operations from left to right.
4. Finally, evaluate any addition and subtraction operations from left to right.

Q: Can you provide more examples of evaluating expressions with absolute value?

A: Yes, here are a few more examples:

• Evaluate the expression $|2x-4|$.
• Evaluate the expression $|3x+2|$.
• Evaluate the expression $|4x-2|$.

Q: How do I simplify expressions with radicals?

A: To simplify expressions with radicals, follow these steps:

1. Simplify any terms that have radicals.
2. Combine like terms.
3. Simplify any exponential expressions.
4. Evaluate any multiplication and division operations from left to right.
5. Finally, evaluate any addition and subtraction operations from left to right.

Q: Can you provide more examples of simplifying expressions with radicals?

A: Yes, here are a few more examples:

• Simplify the expression $\sqrt{16} + \sqrt{9}$.
• Simplify the expression $\sqrt{25} - \sqrt{4}$.
• Simplify the expression $\sqrt{36} + \sqrt{9}$.

Q: How do I evaluate expressions with square roots?

A: To evaluate expressions with square roots, follow these steps:

1. Evaluate any terms that have square roots.
2. Simplify any terms that have square roots.
3. Evaluate any multiplication and division operations from left to right.
4. Finally, evaluate any addition and subtraction operations from left to right.

Q: Can you provide more examples of evaluating expressions with square roots?

A: Yes, here are a few more examples:

• Evaluate the expression $\sqrt{16}$.
• Evaluate the expression $\sqrt{9}$.
• Evaluate the expression $\sqrt{25}$.

Q: How do I simplify expressions with cube roots?

A: To simplify expressions with cube roots, follow these steps:

1. Evaluate any terms that have cube roots.
2. Simplify any terms that have cube roots.
3. Evaluate any multiplication and division operations from left to right.
4. Finally, evaluate any addition and subtraction operations from left to right.

Q: Can you provide more examples of simplifying expressions with cube roots?

A: Yes, here are a few more examples:

• Simplify the expression $\sqrt[3]{27}$.
• Simplify the expression $\sqrt[3]{8}$.
• Simplify the expression $\sqrt[3]{64}$.

Q: How do I evaluate expressions with exponents and radicals?

A: To evaluate expressions with exponents and radicals, follow these steps:

1. Evaluate any exponential expressions.
2. Simplify any terms that have radicals.
3. Combine like terms.
4. Simplify any exponential expressions.
5. Evaluate any multiplication and division operations from left to right.
6. Finally, evaluate any addition and subtraction operations from left to right.

Q: Can you provide more examples of evaluating expressions with exponents and radicals?

A: Yes, here are a few more examples:

• Evaluate the expression $2^3 \sqrt{16}$.
• Evaluate the expression $3^2 \sqrt{9}$.
• Evaluate the expression $4^2 \sqrt{25}$.

Q: How do I simplify expressions with fractions and radicals?

A: To simplify expressions with fractions and radicals, follow these steps:

1. Simplify any terms that have fractions.
2. Simplify any terms that have radicals.
3. Combine like terms.
4. Simplify any exponential expressions.
5. Evaluate any multiplication and division operations from left to right.
6. Finally, evaluate any addition and subtraction operations from left to right.

Q: Can you provide more examples of simplifying expressions with fractions and radicals?

A: Yes, here are a few more examples:

• Simplify the expression $\frac{1}{2} \sqrt{16}$.
• Simplify the expression $\frac{3}{4} \sqrt{9}$.
• Simplify the expression $\frac