Certainly! Let's factor each given polynomial completely step-by-step.
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### Problem 91: [tex]\(-18x - 27\)[/tex]
To factor the polynomial [tex]\(-18x - 27\)[/tex]:
1. Look for the greatest common factor (GCF) of the terms:
- The terms are [tex]\(-18x\)[/tex] and [tex]\(-27\)[/tex].
- The GCF of [tex]\(-18\)[/tex] and [tex]\(-27\)[/tex] is [tex]\(-9\)[/tex].
2. Factor out the GCF:
- [tex]\(-18x - 27\)[/tex] can be written as [tex]\(-9(2x + 3)\)[/tex].
So, the factored form is [tex]\(-9(2x + 3)\)[/tex].
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### Problem 92: [tex]\(x^2 - 100\)[/tex]
To factor the polynomial [tex]\(x^2 - 100\)[/tex]:
1. Recognize it as a difference of squares:
- The general formula for the difference of squares is [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].
2. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- Here, [tex]\(a = x\)[/tex] and [tex]\(b = 10\)[/tex], since [tex]\(100 = 10^2\)[/tex].
3. Apply the difference of squares formula:
- [tex]\(x^2 - 100 = (x - 10)(x + 10)\)[/tex].
So, the factored form is [tex]\((x - 10)(x + 10)\)[/tex].
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### Problem 93: [tex]\(9(-2x - 3)\)[/tex]
To factor the polynomial [tex]\(9(-2x - 3)\)[/tex]:
1. Rearrange if necessary and identify the expression within the parentheses:
- Recognize that [tex]\(-2x - 3\)[/tex] is already a simple binomial which can't be factored further directly.
2. Combine the entire expression:
- The expression [tex]\(9(-2x - 3)\)[/tex] is already factored, with [tex]\(9\)[/tex] as a separate factor and [tex]\(-2x - 3\)[/tex] in parentheses.
Since no further factoring is needed, the factor form remains [tex]\(9(-2x - 3)\)[/tex].
However, it's also correct to write:
[tex]\[ 9(-2x - 3) = -9(2x + 3).\][/tex]
So, the factored form is [tex]\(-9(2x + 3)\)[/tex].
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Summarizing the results:
1. [tex]\(-18x - 27\)[/tex] factors to [tex]\(-9(2x + 3)\)[/tex].
2. [tex]\(x^2 - 100\)[/tex] factors to [tex]\((x - 10)(x + 10)\)[/tex].
3. [tex]\(9(-2x - 3)\)[/tex] factors to [tex]\(-9(2x + 3)\)[/tex].
