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Answer :
To simplify the expression [tex]\((2x - 3)(5x^4 - 7x^3 + 6x^2 - 9)\)[/tex], we'll use the distributive property, which involves multiplying each term in the first polynomial by each term in the second polynomial.
Here's a step-by-step breakdown:
1. Distribute [tex]\(2x\)[/tex] over each term in the polynomial [tex]\(5x^4 - 7x^3 + 6x^2 - 9\)[/tex]:
- [tex]\(2x \cdot 5x^4 = 10x^5\)[/tex]
- [tex]\(2x \cdot (-7x^3) = -14x^4\)[/tex]
- [tex]\(2x \cdot 6x^2 = 12x^3\)[/tex]
- [tex]\(2x \cdot (-9) = -18x\)[/tex]
2. Distribute [tex]\(-3\)[/tex] over each term in the polynomial [tex]\(5x^4 - 7x^3 + 6x^2 - 9\)[/tex]:
- [tex]\(-3 \cdot 5x^4 = -15x^4\)[/tex]
- [tex]\(-3 \cdot (-7x^3) = 21x^3\)[/tex]
- [tex]\(-3 \cdot 6x^2 = -18x^2\)[/tex]
- [tex]\(-3 \cdot (-9) = 27\)[/tex]
3. Combine all the resulting terms:
[tex]\[
10x^5 + (-14x^4) + 12x^3 + (-18x) + (-15x^4) + 21x^3 + (-18x^2) + 27
\][/tex]
4. Combine like terms:
- For [tex]\(x^5\)[/tex]: [tex]\(10x^5\)[/tex]
- For [tex]\(x^4\)[/tex]: [tex]\(-14x^4 + (-15x^4) = -29x^4\)[/tex]
- For [tex]\(x^3\)[/tex]: [tex]\(12x^3 + 21x^3 = 33x^3\)[/tex]
- For [tex]\(x^2\)[/tex]: [tex]\(-18x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(-18x\)[/tex]
- Constant term: [tex]\(27\)[/tex]
5. Therefore, the simplified expression is:
[tex]\[
10x^5 - 29x^4 + 33x^3 - 18x^2 - 18x + 27
\][/tex]
This simplified expression matches the first option in the given choices.
Here's a step-by-step breakdown:
1. Distribute [tex]\(2x\)[/tex] over each term in the polynomial [tex]\(5x^4 - 7x^3 + 6x^2 - 9\)[/tex]:
- [tex]\(2x \cdot 5x^4 = 10x^5\)[/tex]
- [tex]\(2x \cdot (-7x^3) = -14x^4\)[/tex]
- [tex]\(2x \cdot 6x^2 = 12x^3\)[/tex]
- [tex]\(2x \cdot (-9) = -18x\)[/tex]
2. Distribute [tex]\(-3\)[/tex] over each term in the polynomial [tex]\(5x^4 - 7x^3 + 6x^2 - 9\)[/tex]:
- [tex]\(-3 \cdot 5x^4 = -15x^4\)[/tex]
- [tex]\(-3 \cdot (-7x^3) = 21x^3\)[/tex]
- [tex]\(-3 \cdot 6x^2 = -18x^2\)[/tex]
- [tex]\(-3 \cdot (-9) = 27\)[/tex]
3. Combine all the resulting terms:
[tex]\[
10x^5 + (-14x^4) + 12x^3 + (-18x) + (-15x^4) + 21x^3 + (-18x^2) + 27
\][/tex]
4. Combine like terms:
- For [tex]\(x^5\)[/tex]: [tex]\(10x^5\)[/tex]
- For [tex]\(x^4\)[/tex]: [tex]\(-14x^4 + (-15x^4) = -29x^4\)[/tex]
- For [tex]\(x^3\)[/tex]: [tex]\(12x^3 + 21x^3 = 33x^3\)[/tex]
- For [tex]\(x^2\)[/tex]: [tex]\(-18x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(-18x\)[/tex]
- Constant term: [tex]\(27\)[/tex]
5. Therefore, the simplified expression is:
[tex]\[
10x^5 - 29x^4 + 33x^3 - 18x^2 - 18x + 27
\][/tex]
This simplified expression matches the first option in the given choices.
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