Answer :
To find the product of the expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we will follow these steps:
1. Expand the Expression:
We need to multiply the three expressions together. Start by multiplying any two expressions first and then multiply the result by the third expression.
2. Multiply the First Two Terms:
- Multiply [tex]\(7x^2\)[/tex] with each term in the expression [tex]\((2x^3 + 5)\)[/tex].
[tex]\[
(7x^2) \times (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
- This gives us:
[tex]\[
14x^5 + 35x^2
\][/tex]
3. Multiply the Result with the Third Term:
- Now multiply [tex]\((14x^5 + 35x^2)\)[/tex] with [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
(14x^5 + 35x^2) \times (x^2 - 4x - 9)
\][/tex]
- Distribute each term in the first expression across all terms in the second expression.
[tex]\[
14x^5 \cdot x^2 + 14x^5 \cdot (-4x) + 14x^5 \cdot (-9) + 35x^2 \cdot x^2 + 35x^2 \cdot (-4x) + 35x^2 \cdot (-9)
\][/tex]
4. Performing all Multiplications:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
5. Combine Like Terms:
Since all terms are already unique in degrees, no terms to combine further.
6. Final Expression:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the expanded form of the given product expression.
1. Expand the Expression:
We need to multiply the three expressions together. Start by multiplying any two expressions first and then multiply the result by the third expression.
2. Multiply the First Two Terms:
- Multiply [tex]\(7x^2\)[/tex] with each term in the expression [tex]\((2x^3 + 5)\)[/tex].
[tex]\[
(7x^2) \times (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
- This gives us:
[tex]\[
14x^5 + 35x^2
\][/tex]
3. Multiply the Result with the Third Term:
- Now multiply [tex]\((14x^5 + 35x^2)\)[/tex] with [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
(14x^5 + 35x^2) \times (x^2 - 4x - 9)
\][/tex]
- Distribute each term in the first expression across all terms in the second expression.
[tex]\[
14x^5 \cdot x^2 + 14x^5 \cdot (-4x) + 14x^5 \cdot (-9) + 35x^2 \cdot x^2 + 35x^2 \cdot (-4x) + 35x^2 \cdot (-9)
\][/tex]
4. Performing all Multiplications:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
5. Combine Like Terms:
Since all terms are already unique in degrees, no terms to combine further.
6. Final Expression:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the expanded form of the given product expression.
