Let's simplify the product [tex]\((b - 2c)(-3b + c)\)[/tex] step by step.
First, distribute each term in the first binomial to every term in the second binomial:
[tex]\[
(b - 2c)(-3b + c)
\][/tex]
Using the distributive property, this expands to:
[tex]\[
b \cdot (-3b) + b \cdot c - 2c \cdot (-3b) - 2c \cdot c
\][/tex]
Now, calculate each individual term:
[tex]\[
b \cdot (-3b) = -3b^2
\][/tex]
[tex]\[
b \cdot c = bc
\][/tex]
[tex]\[
-2c \cdot (-3b) = 6bc
\][/tex]
[tex]\[
-2c \cdot c = -2c^2
\][/tex]
So the expression is now:
[tex]\[
-3b^2 + bc + 6bc - 2c^2
\][/tex]
Combine like terms:
[tex]\[
-3b^2 + (bc + 6bc) - 2c^2
\][/tex]
[tex]\[
-3b^2 + 7bc - 2c^2
\][/tex]
Now, let's analyze the simplified product [tex]\(-3b^2 + 7bc - 2c^2\)[/tex]:
1. Number of terms: The simplified product has three terms: [tex]\(-3b^2\)[/tex], [tex]\(7bc\)[/tex], and [tex]\(-2c^2\)[/tex].
2. Degree of the polynomial: The highest degree of any term in the polynomial is 2, as the terms are [tex]\(-3b^2\)[/tex], which has a degree of 2, [tex]\(7bc\)[/tex], which has a degree of 2 (since [tex]\(b\)[/tex] and [tex]\(c\)[/tex] are both to the power of 1 and their sum is 2), and [tex]\(-2c^2\)[/tex], which has a degree of 2.
3. Number of negative terms: The polynomial has two negative terms: [tex]\(-3b^2\)[/tex] and [tex]\(-2c^2\)[/tex].
Given this analysis, the correct statements are:
- The simplified product has a degree of 2.
- The simplified product, in standard form, has exactly 2 negative terms.
