Answer :
To find the product of the given expression [tex]\( \left(7x^2\right)\left(2x^3 + 5\right)\left(x^2 - 4x - 9\right) \)[/tex], we'll go through the following steps:
1. Multiply the first expression by the second expression: [tex]\( 7x^2 \cdot (2x^3 + 5) \)[/tex].
2. Distribute [tex]\( 7x^2 \)[/tex] to each term inside the parentheses:
- [tex]\( 7x^2 \cdot 2x^3 = 14x^5 \)[/tex]
- [tex]\( 7x^2 \cdot 5 = 35x^2 \)[/tex]
This gives us the resulting polynomial: [tex]\( 14x^5 + 35x^2 \)[/tex].
3. Multiply the result from step 2 with the third expression: [tex]\( (14x^5 + 35x^2) \cdot (x^2 - 4x - 9) \)[/tex].
4. Distribute each term of [tex]\( (14x^5 + 35x^2) \)[/tex] with each term of [tex]\( (x^2 - 4x - 9) \)[/tex]:
- Multiply [tex]\( 14x^5 \)[/tex] by each term in [tex]\( x^2 - 4x - 9 \)[/tex]:
- [tex]\( 14x^5 \cdot x^2 = 14x^7 \)[/tex]
- [tex]\( 14x^5 \cdot (-4x) = -56x^6 \)[/tex]
- [tex]\( 14x^5 \cdot (-9) = -126x^5 \)[/tex]
- Multiply [tex]\( 35x^2 \)[/tex] by each term in [tex]\( x^2 - 4x - 9 \)[/tex]:
- [tex]\( 35x^2 \cdot x^2 = 35x^4 \)[/tex]
- [tex]\( 35x^2 \cdot (-4x) = -140x^3 \)[/tex]
- [tex]\( 35x^2 \cdot (-9) = -315x^2 \)[/tex]
5. Combine all the terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
So, the product of the three expressions is:
[tex]\[ 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2 \][/tex]
1. Multiply the first expression by the second expression: [tex]\( 7x^2 \cdot (2x^3 + 5) \)[/tex].
2. Distribute [tex]\( 7x^2 \)[/tex] to each term inside the parentheses:
- [tex]\( 7x^2 \cdot 2x^3 = 14x^5 \)[/tex]
- [tex]\( 7x^2 \cdot 5 = 35x^2 \)[/tex]
This gives us the resulting polynomial: [tex]\( 14x^5 + 35x^2 \)[/tex].
3. Multiply the result from step 2 with the third expression: [tex]\( (14x^5 + 35x^2) \cdot (x^2 - 4x - 9) \)[/tex].
4. Distribute each term of [tex]\( (14x^5 + 35x^2) \)[/tex] with each term of [tex]\( (x^2 - 4x - 9) \)[/tex]:
- Multiply [tex]\( 14x^5 \)[/tex] by each term in [tex]\( x^2 - 4x - 9 \)[/tex]:
- [tex]\( 14x^5 \cdot x^2 = 14x^7 \)[/tex]
- [tex]\( 14x^5 \cdot (-4x) = -56x^6 \)[/tex]
- [tex]\( 14x^5 \cdot (-9) = -126x^5 \)[/tex]
- Multiply [tex]\( 35x^2 \)[/tex] by each term in [tex]\( x^2 - 4x - 9 \)[/tex]:
- [tex]\( 35x^2 \cdot x^2 = 35x^4 \)[/tex]
- [tex]\( 35x^2 \cdot (-4x) = -140x^3 \)[/tex]
- [tex]\( 35x^2 \cdot (-9) = -315x^2 \)[/tex]
5. Combine all the terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
So, the product of the three expressions is:
[tex]\[ 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2 \][/tex]
