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Answer :
To determine which of the given options are factors of the expression [tex]\(44x^2y + 44x^2\)[/tex], let's break it down step by step.
1. Factor the Expression:
- The given expression is [tex]\(44x^2y + 44x^2\)[/tex].
- Notice that both terms in the expression have a common factor of [tex]\(44x^2\)[/tex].
- Factor out [tex]\(44x^2\)[/tex] from the expression:
[tex]\[
44x^2y + 44x^2 = 44x^2(y + 1)
\][/tex]
2. Identify Factors of the Factored Expression:
- The expression [tex]\(44x^2(y + 1)\)[/tex] can be broken down into its factors:
- [tex]\(44x^2\)[/tex] is a factor.
- [tex]\(y + 1\)[/tex] is a factor.
- From [tex]\(44x^2\)[/tex], we can further derive its factors: [tex]\(44\)[/tex], [tex]\(x\)[/tex], and [tex]\(x\)[/tex].
- [tex]\(44\)[/tex] itself can be factored into smaller numbers, but it is not necessary for this task.
3. Check the Given Options:
- Here are the options given in the problem:
1. [tex]\(11y + 11\)[/tex]
2. [tex]\(x\)[/tex]
3. [tex]\(11y\)[/tex]
4. [tex]\(3\)[/tex]
5. [tex]\(4\)[/tex]
4. Determine Which are Factors:
- Option 1: [tex]\(11y + 11\)[/tex]
- Simplify: [tex]\(11(y + 1)\)[/tex]
- This matches with the factor [tex]\(y + 1\)[/tex] from the expression, multiplied by 11. However, [tex]\(11(y + 1)\)[/tex] itself was not a direct factor found, so this is not a factor of [tex]\(44x^2(y + 1)\)[/tex].
- Option 2: [tex]\(x\)[/tex]
- Since [tex]\(x\)[/tex] is a component of [tex]\(44x^2\)[/tex], it is indeed a factor.
- Option 3: [tex]\(11y\)[/tex]
- [tex]\(11y\)[/tex] is not part of either [tex]\(44x^2\)[/tex] or [tex]\(y + 1\)[/tex], so it is not a factor.
- Option 4: [tex]\(3\)[/tex]
- While [tex]\(3\)[/tex] is a factor of [tex]\(44\)[/tex], it's not a factor of [tex]\(44x^2\)[/tex] or [tex]\(y + 1\)[/tex] directly in their entirety, hence not a direct factor.
- Option 5: [tex]\(4\)[/tex]
- [tex]\(4\)[/tex] is a factor of [tex]\(44\)[/tex], and since [tex]\(44x^2 = 4 \cdot 11 \cdot x^2\)[/tex], it is a factor.
Based on this detailed analysis, the factors of [tex]\(44x^2y + 44x^2\)[/tex] from the given options are [tex]\(x\)[/tex] and [tex]\(4\)[/tex].
1. Factor the Expression:
- The given expression is [tex]\(44x^2y + 44x^2\)[/tex].
- Notice that both terms in the expression have a common factor of [tex]\(44x^2\)[/tex].
- Factor out [tex]\(44x^2\)[/tex] from the expression:
[tex]\[
44x^2y + 44x^2 = 44x^2(y + 1)
\][/tex]
2. Identify Factors of the Factored Expression:
- The expression [tex]\(44x^2(y + 1)\)[/tex] can be broken down into its factors:
- [tex]\(44x^2\)[/tex] is a factor.
- [tex]\(y + 1\)[/tex] is a factor.
- From [tex]\(44x^2\)[/tex], we can further derive its factors: [tex]\(44\)[/tex], [tex]\(x\)[/tex], and [tex]\(x\)[/tex].
- [tex]\(44\)[/tex] itself can be factored into smaller numbers, but it is not necessary for this task.
3. Check the Given Options:
- Here are the options given in the problem:
1. [tex]\(11y + 11\)[/tex]
2. [tex]\(x\)[/tex]
3. [tex]\(11y\)[/tex]
4. [tex]\(3\)[/tex]
5. [tex]\(4\)[/tex]
4. Determine Which are Factors:
- Option 1: [tex]\(11y + 11\)[/tex]
- Simplify: [tex]\(11(y + 1)\)[/tex]
- This matches with the factor [tex]\(y + 1\)[/tex] from the expression, multiplied by 11. However, [tex]\(11(y + 1)\)[/tex] itself was not a direct factor found, so this is not a factor of [tex]\(44x^2(y + 1)\)[/tex].
- Option 2: [tex]\(x\)[/tex]
- Since [tex]\(x\)[/tex] is a component of [tex]\(44x^2\)[/tex], it is indeed a factor.
- Option 3: [tex]\(11y\)[/tex]
- [tex]\(11y\)[/tex] is not part of either [tex]\(44x^2\)[/tex] or [tex]\(y + 1\)[/tex], so it is not a factor.
- Option 4: [tex]\(3\)[/tex]
- While [tex]\(3\)[/tex] is a factor of [tex]\(44\)[/tex], it's not a factor of [tex]\(44x^2\)[/tex] or [tex]\(y + 1\)[/tex] directly in their entirety, hence not a direct factor.
- Option 5: [tex]\(4\)[/tex]
- [tex]\(4\)[/tex] is a factor of [tex]\(44\)[/tex], and since [tex]\(44x^2 = 4 \cdot 11 \cdot x^2\)[/tex], it is a factor.
Based on this detailed analysis, the factors of [tex]\(44x^2y + 44x^2\)[/tex] from the given options are [tex]\(x\)[/tex] and [tex]\(4\)[/tex].
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