To factor the polynomial [tex]\( 36 p^5 q + 63 p^4 q^3 + 45 p^2 q^2 \)[/tex] and find the sum of the coefficients of all terms inside the parenthesis, follow these detailed steps:
1. Identify the Greatest Common Factor (GCF):
First, determine the GCF of the coefficients and the variables:
- Coefficients: The GCF of 36, 63, and 45 is 9.
- Variables: Each term has at least one [tex]\( p^2 \)[/tex] and one [tex]\( q \)[/tex] as factors.
Therefore, the GCF of the entire polynomial is [tex]\( 9 p^2 q \)[/tex].
2. Factor out the GCF:
Divide each term of the polynomial by [tex]\( 9 p^2 q \)[/tex]:
[tex]\[
\begin{array}{rl}
\frac{36 p^5 q}{9 p^2 q} & = \frac{36}{9} p^{5-2} q^{1-1} = 4 p^3 \\
\frac{63 p^4 q^3}{9 p^2 q} & = \frac{63}{9} p^{4-2} q^{3-1} = 7 p^2 q^2 \\
\frac{45 p^2 q^2}{9 p^2 q} & = \frac{45}{9} p^{2-2} q^{2-1} = 5 q
\end{array}
\][/tex]
After factoring out [tex]\( 9 p^2 q \)[/tex], the polynomial can be written as:
[tex]\[
36 p^5 q + 63 p^4 q^3 + 45 p^2 q^2 = 9 p^2 q (4 p^3 + 7 p^2 q^2 + 5 q)
\][/tex]
3. Sum the coefficients inside the parenthesis:
The factorized polynomial inside the parenthesis is:
[tex]\[
4 p^3 + 7 p^2 q^2 + 5 q
\][/tex]
The coefficients are 4, 7, and 5. Their sum is:
[tex]\[
4 + 7 + 5 = 16
\][/tex]
So, the fully factored form of the polynomial is:
[tex]\[
9 p^2 q (4 p^3 + 7 p^2 q^2 + 5 q)
\][/tex]
And the sum of the coefficients of all terms inside the parenthesis is 16.
Thus, the final factorized polynomial and the sum of the coefficients inside the parenthesis are:
[tex]\[
(9 p^2 q (4 p^3 + 7 p^2 q^2 + 5 q), 16)
\][/tex]
