Answer :
Sure, let's analyze each part of the question one by one and find the coefficients you need.
(a) Coefficient of [tex]\(a^2\)[/tex] in [tex]\(15a^3b^2c^5\)[/tex]:
First, we observe the term [tex]\(15a^3b^2c^5\)[/tex]. The exponent of [tex]\(a\)[/tex] is 3, which means there is no [tex]\(a^2\)[/tex] term specifically isolated in this expression. The overall coefficient of the term [tex]\(15a^3b^2c^5\)[/tex] is 15. Since there is no separate [tex]\(a^2\)[/tex] term within the expression, we consider this overall coefficient. Hence, the coefficient of [tex]\(a^2\)[/tex] is:
[tex]\[ \boxed{15} \][/tex]
(b) Coefficient of [tex]\(b\)[/tex] in [tex]\(\frac{8}{11}a^5b^5\)[/tex]:
Next, consider the term [tex]\(\frac{8}{11}a^5b^5\)[/tex]. The exponent of [tex]\(b\)[/tex] in this term is 5. This means that if we focus on the coefficient of [tex]\(b\)[/tex], we need to extract the part that [tex]\(b\)[/tex] directly relates to, which is [tex]\(b^5\)[/tex]'s coefficient.
In the expression [tex]\(\frac{8}{11}a^5b^5\)[/tex], the coefficient of [tex]\(b\)[/tex] can be considered as [tex]\(\frac{8}{11}\)[/tex]. Hence, the coefficient of [tex]\(b\)[/tex] is:
[tex]\[ \boxed{\frac{8}{11}} \][/tex]
(c) Coefficient of [tex]\(x\)[/tex] in [tex]\(12xy^6z\)[/tex]:
Next, observe the term [tex]\(12xy^6z\)[/tex]. The exponent of [tex]\(x\)[/tex] is 1, which makes it straightforward. The coefficient of [tex]\(x\)[/tex] in this term is the numerical coefficient that is multiplied by [tex]\(x\)[/tex], which in this case is 12.
So, the coefficient of [tex]\(x\)[/tex] is:
[tex]\[ \boxed{12} \][/tex]
(d) Coefficient of [tex]\(q\)[/tex] in [tex]\(p^2q - pq^3 - 2pq + 3\)[/tex]:
Here, let's break down the expression [tex]\(p^2q - pq^3 - 2pq + 3\)[/tex]:
- In the term [tex]\(p^2q\)[/tex], the coefficient of [tex]\(q\)[/tex] is [tex]\(+1 \times p^2\)[/tex].
- In the term [tex]\(-pq^3\)[/tex], the coefficient of [tex]\(q\)[/tex] is not relevant because of the exponent being 3.
- In the term [tex]\(-2pq\)[/tex], the coefficient of [tex]\(q\)[/tex] is [tex]\(-2 \times p\)[/tex].
- The constant term [tex]\(3\)[/tex] does not contribute to the coefficient of [tex]\(q\)[/tex].
When combining the relevant terms for just the coefficient of [tex]\(q\)[/tex], you get:
[tex]\[ p^2 \text{ from } p^2q \text{ and } -2p \text{ from } -2pq \][/tex]
Combining them:
[tex]\[ p^2 + (-2p) \][/tex]
The coefficient of [tex]\(q\)[/tex] when considered collectively and simplified gives us:
- For [tex]\(pq\)[/tex], overall coefficient is [tex]\(-2\)[/tex]
- For [tex]\(pq^3\)[/tex], overall coefficient is [tex]\(-1\)[/tex]
Therefore, the coefficient of [tex]\(q\)[/tex] is:
[tex]\[ \boxed{-2} \][/tex]
[tex]\[ \boxed{-1 \text{ for } pq^3} \][/tex]
(a) Coefficient of [tex]\(a^2\)[/tex] in [tex]\(15a^3b^2c^5\)[/tex]:
First, we observe the term [tex]\(15a^3b^2c^5\)[/tex]. The exponent of [tex]\(a\)[/tex] is 3, which means there is no [tex]\(a^2\)[/tex] term specifically isolated in this expression. The overall coefficient of the term [tex]\(15a^3b^2c^5\)[/tex] is 15. Since there is no separate [tex]\(a^2\)[/tex] term within the expression, we consider this overall coefficient. Hence, the coefficient of [tex]\(a^2\)[/tex] is:
[tex]\[ \boxed{15} \][/tex]
(b) Coefficient of [tex]\(b\)[/tex] in [tex]\(\frac{8}{11}a^5b^5\)[/tex]:
Next, consider the term [tex]\(\frac{8}{11}a^5b^5\)[/tex]. The exponent of [tex]\(b\)[/tex] in this term is 5. This means that if we focus on the coefficient of [tex]\(b\)[/tex], we need to extract the part that [tex]\(b\)[/tex] directly relates to, which is [tex]\(b^5\)[/tex]'s coefficient.
In the expression [tex]\(\frac{8}{11}a^5b^5\)[/tex], the coefficient of [tex]\(b\)[/tex] can be considered as [tex]\(\frac{8}{11}\)[/tex]. Hence, the coefficient of [tex]\(b\)[/tex] is:
[tex]\[ \boxed{\frac{8}{11}} \][/tex]
(c) Coefficient of [tex]\(x\)[/tex] in [tex]\(12xy^6z\)[/tex]:
Next, observe the term [tex]\(12xy^6z\)[/tex]. The exponent of [tex]\(x\)[/tex] is 1, which makes it straightforward. The coefficient of [tex]\(x\)[/tex] in this term is the numerical coefficient that is multiplied by [tex]\(x\)[/tex], which in this case is 12.
So, the coefficient of [tex]\(x\)[/tex] is:
[tex]\[ \boxed{12} \][/tex]
(d) Coefficient of [tex]\(q\)[/tex] in [tex]\(p^2q - pq^3 - 2pq + 3\)[/tex]:
Here, let's break down the expression [tex]\(p^2q - pq^3 - 2pq + 3\)[/tex]:
- In the term [tex]\(p^2q\)[/tex], the coefficient of [tex]\(q\)[/tex] is [tex]\(+1 \times p^2\)[/tex].
- In the term [tex]\(-pq^3\)[/tex], the coefficient of [tex]\(q\)[/tex] is not relevant because of the exponent being 3.
- In the term [tex]\(-2pq\)[/tex], the coefficient of [tex]\(q\)[/tex] is [tex]\(-2 \times p\)[/tex].
- The constant term [tex]\(3\)[/tex] does not contribute to the coefficient of [tex]\(q\)[/tex].
When combining the relevant terms for just the coefficient of [tex]\(q\)[/tex], you get:
[tex]\[ p^2 \text{ from } p^2q \text{ and } -2p \text{ from } -2pq \][/tex]
Combining them:
[tex]\[ p^2 + (-2p) \][/tex]
The coefficient of [tex]\(q\)[/tex] when considered collectively and simplified gives us:
- For [tex]\(pq\)[/tex], overall coefficient is [tex]\(-2\)[/tex]
- For [tex]\(pq^3\)[/tex], overall coefficient is [tex]\(-1\)[/tex]
Therefore, the coefficient of [tex]\(q\)[/tex] is:
[tex]\[ \boxed{-2} \][/tex]
[tex]\[ \boxed{-1 \text{ for } pq^3} \][/tex]
