Answer :
To find the product of the expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we will multiply these terms together step-by-step.
1. Multiply the first two expressions: [tex]\((7x^2)\)[/tex] and [tex]\((2x^3 + 5)\)[/tex].
- Distribute [tex]\(7x^2\)[/tex] to each term in the second expression:
- [tex]\(7x^2 \times 2x^3 = 14x^5\)[/tex]
- [tex]\(7x^2 \times 5 = 35x^2\)[/tex]
- This gives us: [tex]\(14x^5 + 35x^2\)[/tex].
2. Multiply the result with the third expression: [tex]\((14x^5 + 35x^2)\)[/tex] and [tex]\((x^2 - 4x - 9)\)[/tex].
- Distribute each term in [tex]\((14x^5 + 35x^2)\)[/tex] across [tex]\((x^2 - 4x - 9)\)[/tex]:
- [tex]\(14x^5 \times x^2 = 14x^7\)[/tex]
- [tex]\(14x^5 \times (-4x) = -56x^6\)[/tex]
- [tex]\(14x^5 \times (-9) = -126x^5\)[/tex]
- [tex]\(35x^2 \times x^2 = 35x^4\)[/tex]
- [tex]\(35x^2 \times (-4x) = -140x^3\)[/tex]
- [tex]\(35x^2 \times (-9) = -315x^2\)[/tex]
3. Combine all the terms: Collect all the resulting terms from the distribution:
- [tex]\(14x^7 + (-56x^6) + (-126x^5) + 35x^4 + (-140x^3) + (-315x^2)\)[/tex]
Therefore, the expanded and simplified product of the expression is:
[tex]\[ 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2 \][/tex]
This is the final expression in expanded form.
1. Multiply the first two expressions: [tex]\((7x^2)\)[/tex] and [tex]\((2x^3 + 5)\)[/tex].
- Distribute [tex]\(7x^2\)[/tex] to each term in the second expression:
- [tex]\(7x^2 \times 2x^3 = 14x^5\)[/tex]
- [tex]\(7x^2 \times 5 = 35x^2\)[/tex]
- This gives us: [tex]\(14x^5 + 35x^2\)[/tex].
2. Multiply the result with the third expression: [tex]\((14x^5 + 35x^2)\)[/tex] and [tex]\((x^2 - 4x - 9)\)[/tex].
- Distribute each term in [tex]\((14x^5 + 35x^2)\)[/tex] across [tex]\((x^2 - 4x - 9)\)[/tex]:
- [tex]\(14x^5 \times x^2 = 14x^7\)[/tex]
- [tex]\(14x^5 \times (-4x) = -56x^6\)[/tex]
- [tex]\(14x^5 \times (-9) = -126x^5\)[/tex]
- [tex]\(35x^2 \times x^2 = 35x^4\)[/tex]
- [tex]\(35x^2 \times (-4x) = -140x^3\)[/tex]
- [tex]\(35x^2 \times (-9) = -315x^2\)[/tex]
3. Combine all the terms: Collect all the resulting terms from the distribution:
- [tex]\(14x^7 + (-56x^6) + (-126x^5) + 35x^4 + (-140x^3) + (-315x^2)\)[/tex]
Therefore, the expanded and simplified product of the expression is:
[tex]\[ 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2 \][/tex]
This is the final expression in expanded form.
