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Answer :
To factor the expression [tex]\( x^4 - 625y^4 \)[/tex] completely, we can follow these steps:
1. Recognize the Structure:
The expression [tex]\( x^4 - 625y^4 \)[/tex] is a difference of squares. A difference of squares can be expressed as [tex]\( a^2 - b^2 = (a - b)(a + b) \)[/tex].
2. Identify Squares:
Here, notice that [tex]\( x^4 \)[/tex] is the square of [tex]\( x^2 \)[/tex], and [tex]\( 625y^4 \)[/tex] is the square of [tex]\( 25y^2 \)[/tex]. So, we can write:
[tex]\[
x^4 - 625y^4 = (x^2)^2 - (25y^2)^2
\][/tex]
3. Apply Difference of Squares:
Using the difference of squares formula:
[tex]\[
(x^2)^2 - (25y^2)^2 = (x^2 - 25y^2)(x^2 + 25y^2)
\][/tex]
4. Factor Further if Possible:
Look at the first binomial [tex]\( x^2 - 25y^2 \)[/tex] which is again a difference of squares:
[tex]\[
x^2 - 25y^2 = (x)^2 - (5y)^2 = (x - 5y)(x + 5y)
\][/tex]
5. Final Factored Form:
Now, replace [tex]\( x^2 - 25y^2 \)[/tex] in the expression:
[tex]\[
x^4 - 625y^4 = (x - 5y)(x + 5y)(x^2 + 25y^2)
\][/tex]
So, the complete factorization of [tex]\( x^4 - 625y^4 \)[/tex] is [tex]\((x - 5y)(x + 5y)(x^2 + 25y^2)\)[/tex].
1. Recognize the Structure:
The expression [tex]\( x^4 - 625y^4 \)[/tex] is a difference of squares. A difference of squares can be expressed as [tex]\( a^2 - b^2 = (a - b)(a + b) \)[/tex].
2. Identify Squares:
Here, notice that [tex]\( x^4 \)[/tex] is the square of [tex]\( x^2 \)[/tex], and [tex]\( 625y^4 \)[/tex] is the square of [tex]\( 25y^2 \)[/tex]. So, we can write:
[tex]\[
x^4 - 625y^4 = (x^2)^2 - (25y^2)^2
\][/tex]
3. Apply Difference of Squares:
Using the difference of squares formula:
[tex]\[
(x^2)^2 - (25y^2)^2 = (x^2 - 25y^2)(x^2 + 25y^2)
\][/tex]
4. Factor Further if Possible:
Look at the first binomial [tex]\( x^2 - 25y^2 \)[/tex] which is again a difference of squares:
[tex]\[
x^2 - 25y^2 = (x)^2 - (5y)^2 = (x - 5y)(x + 5y)
\][/tex]
5. Final Factored Form:
Now, replace [tex]\( x^2 - 25y^2 \)[/tex] in the expression:
[tex]\[
x^4 - 625y^4 = (x - 5y)(x + 5y)(x^2 + 25y^2)
\][/tex]
So, the complete factorization of [tex]\( x^4 - 625y^4 \)[/tex] is [tex]\((x - 5y)(x + 5y)(x^2 + 25y^2)\)[/tex].
Thank you for reading the article Factor completely tex x 4 625y 4 tex. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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