Answer :
To find the product of the given expressions [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we'll break it down in a step-by-step manner.
1. Distribute the first term: Start by multiplying [tex]\(7x^2\)[/tex] with each term of the second expression [tex]\((2x^3 + 5)\)[/tex]:
[tex]\((7x^2) \cdot (2x^3) = 14x^5\)[/tex]
[tex]\((7x^2) \cdot 5 = 35x^2\)[/tex]
This gives the expression: [tex]\(14x^5 + 35x^2\)[/tex].
2. Multiply the result by the third term: Now we multiply [tex]\((14x^5 + 35x^2)\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/tex].
- Distribute [tex]\(14x^5\)[/tex] across [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]
- Distribute [tex]\(35x^2\)[/tex] across [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]
3. Combine all the terms: Add all the terms obtained from the multiplication:
[tex]\(14x^7 + (-56x^6) + (-126x^5) + 35x^4 + (-140x^3) + (-315x^2)\)[/tex]
This simplifies to:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
Thus, the final product of the expression is:
[tex]\[ 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2 \][/tex]
1. Distribute the first term: Start by multiplying [tex]\(7x^2\)[/tex] with each term of the second expression [tex]\((2x^3 + 5)\)[/tex]:
[tex]\((7x^2) \cdot (2x^3) = 14x^5\)[/tex]
[tex]\((7x^2) \cdot 5 = 35x^2\)[/tex]
This gives the expression: [tex]\(14x^5 + 35x^2\)[/tex].
2. Multiply the result by the third term: Now we multiply [tex]\((14x^5 + 35x^2)\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/tex].
- Distribute [tex]\(14x^5\)[/tex] across [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]
- Distribute [tex]\(35x^2\)[/tex] across [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]
3. Combine all the terms: Add all the terms obtained from the multiplication:
[tex]\(14x^7 + (-56x^6) + (-126x^5) + 35x^4 + (-140x^3) + (-315x^2)\)[/tex]
This simplifies to:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
Thus, the final product of the expression is:
[tex]\[ 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2 \][/tex]
