Answer :
To solve the problem of finding which polynomial expression factors to [tex]\( x^4(4x-7)(4x+7) \)[/tex], we need to identify the polynomial that, when factored, matches this expression.
First, observe that [tex]\( (4x-7)(4x+7) \)[/tex] is a difference of squares:
[tex]\[
(4x-7)(4x+7) = (4x)^2 - 7^2 = 16x^2 - 49
\][/tex]
So, the expression [tex]\( x^4(4x-7)(4x+7) \)[/tex] when expanded, becomes:
[tex]\[
x^4 \cdot (16x^2 - 49)
\][/tex]
Distribute [tex]\( x^4 \)[/tex]:
[tex]\[
x^4 \cdot 16x^2 - x^4 \cdot 49 = 16x^6 - 49x^4
\][/tex]
Thus, we are looking for the polynomial expression that expands to [tex]\( 16x^6 - 49x^4 \)[/tex].
Now, let's check 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]
From the analysis above, we can see that the correct expression is [tex]\( 16x^6 - 49x^4 \)[/tex], which matches option 3.
Therefore, the correct answer is:
[tex]\( 16x^6 - 49x^4 \)[/tex]
First, observe that [tex]\( (4x-7)(4x+7) \)[/tex] is a difference of squares:
[tex]\[
(4x-7)(4x+7) = (4x)^2 - 7^2 = 16x^2 - 49
\][/tex]
So, the expression [tex]\( x^4(4x-7)(4x+7) \)[/tex] when expanded, becomes:
[tex]\[
x^4 \cdot (16x^2 - 49)
\][/tex]
Distribute [tex]\( x^4 \)[/tex]:
[tex]\[
x^4 \cdot 16x^2 - x^4 \cdot 49 = 16x^6 - 49x^4
\][/tex]
Thus, we are looking for the polynomial expression that expands to [tex]\( 16x^6 - 49x^4 \)[/tex].
Now, let's check 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]
From the analysis above, we can see that the correct expression is [tex]\( 16x^6 - 49x^4 \)[/tex], which matches option 3.
Therefore, the correct answer is:
[tex]\( 16x^6 - 49x^4 \)[/tex]
