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Answer :
To find the polynomial expression that factors to [tex]\(x^4(4x-7)(4x+7)\)[/tex], let's analyze what the factorization means and expand it.
The expression [tex]\(x^4(4x-7)(4x+7)\)[/tex] involves two binomials multiplied together:
1. [tex]\((4x-7)\)[/tex]
2. [tex]\((4x+7)\)[/tex]
Notice that [tex]\((4x-7)(4x+7)\)[/tex] follows the pattern of a "difference of squares," which is generally expressed as:
[tex]\[a^2 - b^2 = (a-b)(a+b)\][/tex]
In this case, [tex]\(a = 4x\)[/tex] and [tex]\(b = 7\)[/tex], so:
[tex]\[(4x-7)(4x+7) = (4x)^2 - 7^2 = 16x^2 - 49\][/tex]
Now, multiply this by [tex]\(x^4\)[/tex]:
[tex]\[x^4(16x^2 - 49) = 16x^6 - 49x^4\][/tex]
Therefore, we are looking for the polynomial expression that factors to [tex]\(16x^6 - 49x^4\)[/tex].
Examining the given options:
1. [tex]\(4x^6 - 7x^4\)[/tex]
2. [tex]\(16x^4 - 49x^2\)[/tex]
3. [tex]\(16x^6 - 49x^4\)[/tex]
4. [tex]\(16x^6 + 49x^4\)[/tex]
The correct expression, which matches [tex]\(16x^6 - 49x^4\)[/tex], is option 3:
[tex]\[16x^6 - 49x^4\][/tex]
So, the correct factorization corresponds to option 3: [tex]\(16x^6 - 49x^4\)[/tex].
The expression [tex]\(x^4(4x-7)(4x+7)\)[/tex] involves two binomials multiplied together:
1. [tex]\((4x-7)\)[/tex]
2. [tex]\((4x+7)\)[/tex]
Notice that [tex]\((4x-7)(4x+7)\)[/tex] follows the pattern of a "difference of squares," which is generally expressed as:
[tex]\[a^2 - b^2 = (a-b)(a+b)\][/tex]
In this case, [tex]\(a = 4x\)[/tex] and [tex]\(b = 7\)[/tex], so:
[tex]\[(4x-7)(4x+7) = (4x)^2 - 7^2 = 16x^2 - 49\][/tex]
Now, multiply this by [tex]\(x^4\)[/tex]:
[tex]\[x^4(16x^2 - 49) = 16x^6 - 49x^4\][/tex]
Therefore, we are looking for the polynomial expression that factors to [tex]\(16x^6 - 49x^4\)[/tex].
Examining the given options:
1. [tex]\(4x^6 - 7x^4\)[/tex]
2. [tex]\(16x^4 - 49x^2\)[/tex]
3. [tex]\(16x^6 - 49x^4\)[/tex]
4. [tex]\(16x^6 + 49x^4\)[/tex]
The correct expression, which matches [tex]\(16x^6 - 49x^4\)[/tex], is option 3:
[tex]\[16x^6 - 49x^4\][/tex]
So, the correct factorization corresponds to option 3: [tex]\(16x^6 - 49x^4\)[/tex].
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