Let's match the given polynomial expressions with their corresponding terms. The expressions provided are:
Choices:
a. [tex]\(4 x y^3\)[/tex]
b. [tex]\(14 x y\)[/tex]
c. [tex]\(15 y^2\)[/tex]
d. [tex]\(-6\)[/tex]
e. [tex]\(5 x^2\)[/tex]
Matches:
1. [tex]\(-12 x^2\)[/tex]
2. [tex]\(6 x y\)[/tex]
Our task is to match the expressions from the choices with the matching terms from the matches.
Step-by-step solution:
1. Match Choice [tex]\( a \)[/tex]: [tex]\(4 x y^3\)[/tex]
- This term involves [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(y\)[/tex] raised to the power of 3. None of the matches contain the exact combination of these variables and exponents.
- Therefore, there is no match for [tex]\(4 x y^3\)[/tex].
2. Match Choice [tex]\( b \)[/tex]: [tex]\(14 x y\)[/tex]
- This term involves both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] without any exponent on [tex]\(y\)[/tex], making it a linear term. The match with a similar structure is [tex]\(6 x y\)[/tex].
- Thus, [tex]\(\boxed{b = 2}\)[/tex].
3. Match Choice [tex]\( c \)[/tex]: [tex]\(15 y^2\)[/tex]
- This term involves only [tex]\(y\)[/tex] raised to the power of 2. None of the given matches include a term solely involving [tex]\(y^2\)[/tex].
- Therefore, there is no match for [tex]\(15 y^2\)[/tex].
4. Match Choice [tex]\( d \)[/tex]: [tex]\(-6\)[/tex]
- This term is a constant, as it does not contain any variables ([tex]\(x\)[/tex] or [tex]\(y\)[/tex]). The matches do not include any constants.
- Therefore, there is no match for [tex]\(-6\)[/tex].
5. Match Choice [tex]\( e \)[/tex]: [tex]\(5 x^2\)[/tex]
- This term involves [tex]\(x\)[/tex] squared. We should find a match where [tex]\(x\)[/tex] is squared. The match is [tex]\(-12 x^2\)[/tex].
- Thus, [tex]\(\boxed{e = 1}\)[/tex].
So, the final matches are:
- [tex]\(a = \text{No match}\)[/tex]
- [tex]\(b = 2\)[/tex]
- [tex]\(c = \text{No match}\)[/tex]
- [tex]\(d = \text{No match}\)[/tex]
- [tex]\(e = 1\)[/tex]
We can summarize the matching results as follows:
They form the answer set,
[tex]\[
\mathbf{\{ 'a':\ \text{None},\ 'b':\ '2',\ 'c':\ \text{None},\ 'd':\ \text{None},\ 'e':\ '1' \}}
\][/tex]
This matching process helps us understand which polynomial terms from the choices best align with those provided in the matches based on their structure and components.
