True Statements About The Simplified Product Of (b-2c)(-3b+c)

Hey guys! Let's dive into this math problem together and break it down step by step. We're going to explore the simplified product of the expression $(b-2c)(-3b+c)$ and figure out which statements are true. This involves understanding how to expand the expression, combine like terms, and identify the characteristics of the resulting polynomial. So, grab your thinking caps, and let's get started!

Expanding the Product: A Crucial First Step

To begin, we need to expand the product $(b-2c)(-3b+c)$. This involves using the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). This method ensures that each term in the first binomial is multiplied by each term in the second binomial. Let's break it down:

• First: Multiply the first terms of each binomial: $(b) \cdot (-3b) = -3b^2$
• Outer: Multiply the outer terms of the binomials: $(b) \cdot (c) = bc$
• Inner: Multiply the inner terms of the binomials: $(-2c) \cdot (-3b) = 6bc$
• Last: Multiply the last terms of each binomial: $(-2c) \cdot (c) = -2c^2$

So, after applying the distributive property, we get:

$$-3b^2 + bc + 6bc - 2c^2$$

This expression currently has four terms, but we're not done yet! We need to simplify it further by combining like terms. This step is crucial to understanding the final form of the product and answering the given questions correctly. Let's move on to the next section to see how this is done.

Combining Like Terms: Simplifying the Expression

Now that we've expanded the product, we have the expression $-3b^2 + bc + 6bc - 2c^2$. The next step is to combine the like terms. Like terms are terms that have the same variables raised to the same powers. In this expression, the like terms are $bc$ and $6bc$.

To combine these terms, we simply add their coefficients. The coefficient of $bc$ is 1 (since it's 1 * bc), and the coefficient of $6bc$ is 6. Adding these coefficients gives us $1 + 6 = 7$. Therefore, $bc + 6bc = 7bc$.

Now, let's rewrite the expression with the like terms combined:

$$-3b^2 + 7bc - 2c^2$$

This is the fully simplified form of the product. Notice that we now have three terms: $-3b^2$, $7bc$, and $-2c^2$. This simplified form will help us determine the correct statements in the original question. Let's move on to analyzing the characteristics of this simplified product.

Analyzing the Simplified Product: Terms and Degree

With the expression fully simplified to $-3b^2 + 7bc - 2c^2$, we can now analyze its characteristics to determine which statements are true. Let's consider the number of terms and the degree of the expression.

Number of Terms

A term is a single mathematical expression that is separated from other terms by addition or subtraction. In our simplified expression, we have three terms:

1. $-3b^2$
2. $7bc$
3. $-2c^2$

So, the simplified product has 3 terms. This information helps us evaluate the options given in the original question.

Degree of the Simplified Product

The degree of a term is the sum of the exponents of the variables in that term. The degree of a polynomial is the highest degree of any of its terms. Let's find the degree of each term in our simplified expression:

1. $-3b^2$: The variable is $b$ and its exponent is 2. So, the degree of this term is 2.
2. $7bc$: We have two variables, $b$ and $c$, each with an exponent of 1 (since $b = b^1$ and $c = c^1$). The sum of the exponents is $1 + 1 = 2$. So, the degree of this term is 2.
3. $-2c^2$: The variable is $c$ and its exponent is 2. So, the degree of this term is 2.

Since all the terms have a degree of 2, the degree of the entire simplified product is 2. This is because the degree of a polynomial is the highest degree among its terms.

Now that we know the simplified product has 3 terms and a degree of 2, we can confidently choose the correct statements from the given options. Let's recap the key points and then answer the question.

Recapping Key Points

Before we identify the correct statements, let's quickly recap the steps we took:

1. We expanded the product $(b-2c)(-3b+c)$ using the distributive property (FOIL method), resulting in $-3b^2 + bc + 6bc - 2c^2$.
2. We combined like terms ($bc$ and $6bc$) to simplify the expression to $-3b^2 + 7bc - 2c^2$.
3. We analyzed the simplified product and determined that it has 3 terms and a degree of 2.

With this information, we are now ready to select the true statements from the original options. This systematic approach ensures that we understand each step and can confidently arrive at the correct answer.

Identifying the Correct Statements

Now, let's revisit the original question and the given options:

Which statements are true about the fully simplified product of $(b-2 c)(-3 b+c)$? Select two options.

A. The simplified product has 2 terms. B. The simplified product has 4 terms. C. The simplified product has a degree of 2. D. The simplified product ...

Based on our analysis, we know the following:

• The simplified product has 3 terms.
• The simplified product has a degree of 2.

Therefore, we can conclude that:

• Option A is incorrect because the product has 3 terms, not 2.
• Option B is incorrect because the product has 3 terms, not 4.
• Option C is correct because the product has a degree of 2.

To fully answer the question, we need to select two correct options. Since we've already identified that option C is correct, we need to examine option D. (The full text of option D was not provided in the original prompt, but we can infer it based on the context.)

Let's assume option D states something about the coefficients or the variables in the simplified product. A likely option D could be: The simplified product contains a term with a coefficient of 7. Looking at our simplified product $-3b^2 + 7bc - 2c^2$, we can see that the term $7bc$ has a coefficient of 7. Therefore, option D would also be correct.

So, the two correct statements are:

• C. The simplified product has a degree of 2.
• D. (Assuming) The simplified product contains a term with a coefficient of 7.

Final Thoughts and Tips

Great job, guys! We've successfully unraveled the simplified product of $(b-2c)(-3b+c)$. Remember, the key to solving these types of problems is to:

1. Expand the product carefully using the distributive property (FOIL method).
2. Combine like terms to simplify the expression.
3. Analyze the simplified expression to determine its characteristics, such as the number of terms and the degree.

By following these steps, you can confidently tackle similar math problems. Keep practicing, and you'll become a pro at simplifying algebraic expressions! If you have any questions, feel free to ask. Keep learning and exploring!