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Answer :
Sure! Let's factor the polynomial [tex]\(256 y^4 - 625\)[/tex] completely, step-by-step.
### Step-by-Step Solution
1. Recognize the polynomial structure: The given polynomial is [tex]\(256 y^4 - 625\)[/tex]. It resembles a difference of squares structure because it can be written as [tex]\(a^2 - b^2\)[/tex], where [tex]\(a = (4y^2)\)[/tex] and [tex]\(b = 25\)[/tex].
2. Express as a difference of squares:
[tex]\[
256 y^4 - 625 = (16 y^2)^2 - 25^2
\][/tex]
Here, [tex]\(16y^2\)[/tex] is the square of [tex]\(4y\)[/tex] and [tex]\(625\)[/tex] is the square of [tex]\(25\)[/tex].
3. Apply the difference of squares formula: Recall the difference of squares formula, which is [tex]\(A^2 - B^2 = (A - B)(A + B)\)[/tex]. In our case [tex]\(A = 16y^2\)[/tex] and [tex]\(B = 25\)[/tex], we apply it as follows:
[tex]\[
(16 y^2)^2 - 25^2 = (16 y^2 - 25)(16 y^2 + 25)
\][/tex]
4. Factor [tex]\(16 y^2 - 25\)[/tex]: Notice that [tex]\(16 y^2 - 25\)[/tex] is also a difference of squares. We can factor it further using the difference of squares formula:
[tex]\[
16 y^2 - 25 = (4 y)^2 - 5^2 = (4 y - 5)(4 y + 5)
\][/tex]
5. Put all factors together: Now, we combine all the factored parts:
[tex]\[
256 y^4 - 625 = (4 y - 5)(4 y + 5)(16 y^2 + 25)
\][/tex]
So, the completely factored form of the polynomial [tex]\(256 y^4 - 625\)[/tex] is:
[tex]\[
(4 y - 5)(4 y + 5)(16 y^2 + 25)
\][/tex]
That's the solution!
### Step-by-Step Solution
1. Recognize the polynomial structure: The given polynomial is [tex]\(256 y^4 - 625\)[/tex]. It resembles a difference of squares structure because it can be written as [tex]\(a^2 - b^2\)[/tex], where [tex]\(a = (4y^2)\)[/tex] and [tex]\(b = 25\)[/tex].
2. Express as a difference of squares:
[tex]\[
256 y^4 - 625 = (16 y^2)^2 - 25^2
\][/tex]
Here, [tex]\(16y^2\)[/tex] is the square of [tex]\(4y\)[/tex] and [tex]\(625\)[/tex] is the square of [tex]\(25\)[/tex].
3. Apply the difference of squares formula: Recall the difference of squares formula, which is [tex]\(A^2 - B^2 = (A - B)(A + B)\)[/tex]. In our case [tex]\(A = 16y^2\)[/tex] and [tex]\(B = 25\)[/tex], we apply it as follows:
[tex]\[
(16 y^2)^2 - 25^2 = (16 y^2 - 25)(16 y^2 + 25)
\][/tex]
4. Factor [tex]\(16 y^2 - 25\)[/tex]: Notice that [tex]\(16 y^2 - 25\)[/tex] is also a difference of squares. We can factor it further using the difference of squares formula:
[tex]\[
16 y^2 - 25 = (4 y)^2 - 5^2 = (4 y - 5)(4 y + 5)
\][/tex]
5. Put all factors together: Now, we combine all the factored parts:
[tex]\[
256 y^4 - 625 = (4 y - 5)(4 y + 5)(16 y^2 + 25)
\][/tex]
So, the completely factored form of the polynomial [tex]\(256 y^4 - 625\)[/tex] is:
[tex]\[
(4 y - 5)(4 y + 5)(16 y^2 + 25)
\][/tex]
That's the solution!
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