Certainly! Let's expand and simplify the polynomial expression [tex]\(\left(7 p^4 + 9\right)\left(49 p^8 - 7 p^4 q + q^2\right)\)[/tex].
We need to apply the distributive property (also known as the FOIL method for binomials) to expand this expression.
Step 1: Distribute [tex]\( 7p^4 \)[/tex] in the first polynomial to each term in the second polynomial:
[tex]\[ 7p^4 \cdot (49p^8 - 7p^4q + q^2) \][/tex]
Perform each multiplication:
[tex]\[ 7p^4 \cdot 49p^8 = 343p^{12} \][/tex]
[tex]\[ 7p^4 \cdot (-7p^4q) = -49p^8q \][/tex]
[tex]\[ 7p^4 \cdot q^2 = 7p^4q^2 \][/tex]
So, the distributed result of [tex]\( 7p^4 \)[/tex] is:
[tex]\[ 343p^{12} - 49p^8q + 7p^4q^2 \][/tex]
Step 2: Distribute [tex]\( 9 \)[/tex] in the first polynomial to each term in the second polynomial:
[tex]\[ 9 \cdot (49p^8 - 7p^4q + q^2) \][/tex]
Perform each multiplication:
[tex]\[ 9 \cdot 49p^8 = 441p^8 \][/tex]
[tex]\[ 9 \cdot (-7p^4q) = -63p^4q \][/tex]
[tex]\[ 9 \cdot q^2 = 9q^2 \][/tex]
So, the distributed result of [tex]\( 9 \)[/tex] is:
[tex]\[ 441p^8 - 63p^4q + 9q^2 \][/tex]
Step 3: Combine all the terms from both distributions:
[tex]\[ 343p^{12} - 49p^8q + 7p^4q^2 + 441p^8 - 63p^4q + 9q^2 \][/tex]
Step 4: Group and simplify like terms:
[tex]\[ 343p^{12} + 441p^8 - 49p^8q - 63p^4q + 7p^4q^2 + 9q^2 \][/tex]
There are no further like terms to combine, so the final expanded form of the polynomial is:
[tex]\[ 343p^{12} + 441p^8 - 49p^8q - 63p^4q + 7p^4q^2 + 9q^2 \][/tex]
This is the fully expanded and simplified result of the given polynomial expression.
