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Answer :
To find the product of the expressions [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we'll follow these steps:
1. Distribute the Terms: We'll multiply the expressions step-by-step rather than all at once. First, focus on multiplying the first two expressions, [tex]\((7x^2)\)[/tex] and [tex]\((2x^3 + 5)\)[/tex].
2. First Multiplication:
- Multiply [tex]\(7x^2\)[/tex] by each term in the expression [tex]\(2x^3 + 5\)[/tex].
- [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]
3. Second Multiplication:
- Now multiply the resulting polynomial with the third expression [tex]\((x^2 - 4x - 9)\)[/tex].
4. Distribute Again: Multiply each term from the first polynomial [tex]\(14x^5 + 35x^2\)[/tex] with each term in [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]
5. Combine All Terms: Now, combine all of these results and arrange them in order of decreasing powers of [tex]\(x\)[/tex]:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the expanded form of [tex]\(\left(7x^2\right)\left(2x^3+5\right)\left(x^2-4x-9\right)\)[/tex]. Each step follows the process of distributing terms carefully through polynomial multiplication, resulting in a final polynomial expression.
1. Distribute the Terms: We'll multiply the expressions step-by-step rather than all at once. First, focus on multiplying the first two expressions, [tex]\((7x^2)\)[/tex] and [tex]\((2x^3 + 5)\)[/tex].
2. First Multiplication:
- Multiply [tex]\(7x^2\)[/tex] by each term in the expression [tex]\(2x^3 + 5\)[/tex].
- [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]
3. Second Multiplication:
- Now multiply the resulting polynomial with the third expression [tex]\((x^2 - 4x - 9)\)[/tex].
4. Distribute Again: Multiply each term from the first polynomial [tex]\(14x^5 + 35x^2\)[/tex] with each term in [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]
5. Combine All Terms: Now, combine all of these results and arrange them in order of decreasing powers of [tex]\(x\)[/tex]:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the expanded form of [tex]\(\left(7x^2\right)\left(2x^3+5\right)\left(x^2-4x-9\right)\)[/tex]. Each step follows the process of distributing terms carefully through polynomial multiplication, resulting in a final polynomial expression.
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