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Answer :
To find the product of the given expressions [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we need to multiply them, following these steps:
1. Multiply the first two expressions: Start by multiplying [tex]\(7x^2\)[/tex] with each term in the second expression [tex]\(2x^3 + 5\)[/tex].
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(2x^3\)[/tex]: [tex]\(7x^2 \times 2x^3 = 14x^5\)[/tex].
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(5\)[/tex]: [tex]\(7x^2 \times 5 = 35x^2\)[/tex].
This gives us the intermediate expression: [tex]\(14x^5 + 35x^2\)[/tex].
2. Multiply the result with the third expression: Next, take the intermediate result [tex]\(14x^5 + 35x^2\)[/tex] and multiply it by each term in [tex]\(x^2 - 4x - 9\)[/tex].
- Multiply both terms in [tex]\(14x^5 + 35x^2\)[/tex] by [tex]\(x^2\)[/tex]:
- [tex]\(14x^5 \times x^2 = 14x^7\)[/tex],
- [tex]\(35x^2 \times x^2 = 35x^4\)[/tex].
- Multiply both terms in [tex]\(14x^5 + 35x^2\)[/tex] by [tex]\(-4x\)[/tex]:
- [tex]\(14x^5 \times -4x = -56x^6\)[/tex],
- [tex]\(35x^2 \times -4x = -140x^3\)[/tex].
- Multiply both terms in [tex]\(14x^5 + 35x^2\)[/tex] by [tex]\(-9\)[/tex]:
- [tex]\(14x^5 \times -9 = -126x^5\)[/tex],
- [tex]\(35x^2 \times -9 = -315x^2\)[/tex].
3. Combine and simplify the terms:
- Combine all the terms:
[tex]\(14x^7 + (-56x^6) + (-126x^5) + 35x^4 + (-140x^3) + (-315x^2)\)[/tex].
- Write this as a polynomial in standard form:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2.
\][/tex]
Therefore, the complete expanded form of the product of these expressions is [tex]\[14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2.\][/tex]
1. Multiply the first two expressions: Start by multiplying [tex]\(7x^2\)[/tex] with each term in the second expression [tex]\(2x^3 + 5\)[/tex].
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(2x^3\)[/tex]: [tex]\(7x^2 \times 2x^3 = 14x^5\)[/tex].
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(5\)[/tex]: [tex]\(7x^2 \times 5 = 35x^2\)[/tex].
This gives us the intermediate expression: [tex]\(14x^5 + 35x^2\)[/tex].
2. Multiply the result with the third expression: Next, take the intermediate result [tex]\(14x^5 + 35x^2\)[/tex] and multiply it by each term in [tex]\(x^2 - 4x - 9\)[/tex].
- Multiply both terms in [tex]\(14x^5 + 35x^2\)[/tex] by [tex]\(x^2\)[/tex]:
- [tex]\(14x^5 \times x^2 = 14x^7\)[/tex],
- [tex]\(35x^2 \times x^2 = 35x^4\)[/tex].
- Multiply both terms in [tex]\(14x^5 + 35x^2\)[/tex] by [tex]\(-4x\)[/tex]:
- [tex]\(14x^5 \times -4x = -56x^6\)[/tex],
- [tex]\(35x^2 \times -4x = -140x^3\)[/tex].
- Multiply both terms in [tex]\(14x^5 + 35x^2\)[/tex] by [tex]\(-9\)[/tex]:
- [tex]\(14x^5 \times -9 = -126x^5\)[/tex],
- [tex]\(35x^2 \times -9 = -315x^2\)[/tex].
3. Combine and simplify the terms:
- Combine all the terms:
[tex]\(14x^7 + (-56x^6) + (-126x^5) + 35x^4 + (-140x^3) + (-315x^2)\)[/tex].
- Write this as a polynomial in standard form:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2.
\][/tex]
Therefore, the complete expanded form of the product of these expressions is [tex]\[14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2.\][/tex]
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