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Answer :
To simplify the expression [tex]\((2x - 3)(5x^4 - 7x^3 + 6x^2 - 9)\)[/tex], we’ll multiply the terms using the distributive property. Here’s how you can do it step-by-step:
### Step 1: Expand the expression
Multiply each term in the first binomial by each term in the second polynomial:
1. [tex]\(2x \cdot 5x^4 = 10x^5\)[/tex]
2. [tex]\(2x \cdot (-7x^3) = -14x^4\)[/tex]
3. [tex]\(2x \cdot 6x^2 = 12x^3\)[/tex]
4. [tex]\(2x \cdot (-9) = -18x\)[/tex]
5. [tex]\(-3 \cdot 5x^4 = -15x^4\)[/tex]
6. [tex]\(-3 \cdot (-7x^3) = 21x^3\)[/tex]
7. [tex]\(-3 \cdot 6x^2 = -18x^2\)[/tex]
8. [tex]\(-3 \cdot (-9) = 27\)[/tex]
### Step 2: Combine like terms
Now, add the results of each multiplication:
- [tex]\(10x^5\)[/tex] is the only fifth degree term.
- Combine the fourth degree terms: [tex]\(-14x^4 + (-15x^4) = -29x^4\)[/tex]
- Combine the third degree terms: [tex]\(12x^3 + 21x^3 = 33x^3\)[/tex]
- Combine the second degree terms: [tex]\(-18x^2\)[/tex] (only one)
- Combine the first degree terms: [tex]\(-18x\)[/tex] (only one)
- The constant term: [tex]\(27\)[/tex]
### Simplified Expression
Putting it all together, the simplified expression is:
[tex]\[ 10x^5 - 29x^4 + 33x^3 - 18x^2 - 18x + 27 \][/tex]
Hence, the correct answer is:
[tex]\[ 10x^5 - 29x^4 + 33x^3 - 18x^2 - 18x + 27 \][/tex]
This matches with the provided list of options.
### Step 1: Expand the expression
Multiply each term in the first binomial by each term in the second polynomial:
1. [tex]\(2x \cdot 5x^4 = 10x^5\)[/tex]
2. [tex]\(2x \cdot (-7x^3) = -14x^4\)[/tex]
3. [tex]\(2x \cdot 6x^2 = 12x^3\)[/tex]
4. [tex]\(2x \cdot (-9) = -18x\)[/tex]
5. [tex]\(-3 \cdot 5x^4 = -15x^4\)[/tex]
6. [tex]\(-3 \cdot (-7x^3) = 21x^3\)[/tex]
7. [tex]\(-3 \cdot 6x^2 = -18x^2\)[/tex]
8. [tex]\(-3 \cdot (-9) = 27\)[/tex]
### Step 2: Combine like terms
Now, add the results of each multiplication:
- [tex]\(10x^5\)[/tex] is the only fifth degree term.
- Combine the fourth degree terms: [tex]\(-14x^4 + (-15x^4) = -29x^4\)[/tex]
- Combine the third degree terms: [tex]\(12x^3 + 21x^3 = 33x^3\)[/tex]
- Combine the second degree terms: [tex]\(-18x^2\)[/tex] (only one)
- Combine the first degree terms: [tex]\(-18x\)[/tex] (only one)
- The constant term: [tex]\(27\)[/tex]
### Simplified Expression
Putting it all together, the simplified expression is:
[tex]\[ 10x^5 - 29x^4 + 33x^3 - 18x^2 - 18x + 27 \][/tex]
Hence, the correct answer is:
[tex]\[ 10x^5 - 29x^4 + 33x^3 - 18x^2 - 18x + 27 \][/tex]
This matches with the provided list of options.
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