The Product Of A Binomial And A Trinomial Is X 3 + 3 X 2 − X + 2 X 2 + 6 X − 2 X^3 + 3x^2 - X + 2x^2 + 6x - 2 X 3 + 3 X 2 − X + 2 X 2 + 6 X − 2 . Which Expression Is Equivalent To This Product After It Has Been Fully Simplified?A. X 3 + 5 X 2 + 5 X − 2 X^3 + 5x^2 + 5x - 2 X 3 + 5 X 2 + 5 X − 2 B. X 3 + 2 X 2 + 8 X − 2 X^3 + 2x^2 + 8x - 2 X 3 + 2 X 2 + 8 X − 2 C. $x^3 +

Understanding the Problem

The given problem involves the product of a binomial and a trinomial, resulting in the expression $x^3 + 3x^2 - x + 2x^2 + 6x - 2$. The task is to simplify this expression and determine which of the provided options is equivalent to the simplified form.

Step 1: Combine Like Terms

To simplify the given expression, we need to combine like terms. The expression can be rewritten as:

$x^3 + (3x^2 + 2x^2) + (-x + 6x) - 2$

Combining the like terms in the expression

$x^3 + 5x^2 + 5x - 2$

Step 2: Analyze the Options

Now that we have simplified the expression, let's analyze the provided options to determine which one is equivalent to the simplified form.

Option A: $x^3 + 5x^2 + 5x - 2$

This option matches the simplified expression we obtained in Step 1.

Option B: $x^3 + 2x^2 + 8x - 2$

This option does not match the simplified expression, as the coefficient of $x^2$ is different.

Option C: $x^3 + 5x^2 + 5x - 2$

This option also matches the simplified expression, but it is identical to Option A.

Conclusion

Based on the analysis, we can conclude that the expression equivalent to the product of a binomial and a trinomial after it has been fully simplified is:

$x^3 + 5x^2 + 5x - 2$

This expression matches both Option A and Option C, but since the question asks for a single expression, we can consider either of these options as the correct answer.

Final Answer

Understanding the Problem

The given problem involves the product of a binomial and a trinomial, resulting in the expression $x^3 + 3x^2 - x + 2x^2 + 6x - 2$. The task is to simplify this expression and determine which of the provided options is equivalent to the simplified form.

Q&A Session

Q: What is the product of a binomial and a trinomial?

A: The product of a binomial and a trinomial is a polynomial expression that results from multiplying a binomial (a polynomial with two terms) by a trinomial (a polynomial with three terms).

Q: How do I simplify the given expression?

A: To simplify the given expression, you need to combine like terms. This involves grouping together the terms with the same variable and exponent, and then adding or subtracting the coefficients.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, $3x^2$ and $2x^2$ are like terms because they both have the variable $x$ and the exponent $2$.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms. For example, if you have $3x^2$ and $2x^2$, you can combine them by adding the coefficients: $3x^2 + 2x^2 = 5x^2$.

Q: What is the simplified expression?

A: The simplified expression is $x^3 + 5x^2 + 5x - 2$.

Q: Which option is equivalent to the simplified expression?

A: The options that are equivalent to the simplified expression are Option A: $x^3 + 5x^2 + 5x - 2$ and Option C: $x^3 + 5x^2 + 5x - 2$. Both of these options match the simplified expression.

Q: Why are there two options that match the simplified expression?

A: There are two options that match the simplified expression because they are identical. Option A and Option C are the same expression, so they both match the simplified expression.

Q: What is the final answer?

A: The final answer is $\boxed{A}$, but since Option C is also correct, you can consider either of these options as the correct answer.

Common Mistakes

• Not combining like terms correctly
• Not identifying like terms
• Not adding or subtracting coefficients correctly
• Not checking the options carefully

Tips and Tricks

• Make sure to combine like terms carefully
• Check the options carefully to make sure they match the simplified expression
• Use a calculator or a computer program to check your work
• Practice, practice, practice!

Conclusion

The product of a binomial and a trinomial is a polynomial expression that results from multiplying a binomial by a trinomial. To simplify the given expression, you need to combine like terms. The simplified expression is $x^3 + 5x^2 + 5x - 2$. The options that are equivalent to the simplified expression are Option A: $x^3 + 5x^2 + 5x - 2$ and Option C: $x^3 + 5x^2 + 5x - 2$. The final answer is $\boxed{A}$, but since Option C is also correct, you can consider either of these options as the correct answer.