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Answer :
To factor the polynomial expression [tex]\(16y^4 - 625x^4\)[/tex], we can utilize factorization techniques involving difference of squares. Here's how you can approach the problem step by step:
1. Recognize it as a Difference of Squares:
The expression [tex]\(16y^4 - 625x^4\)[/tex] is in the form of a difference of squares. Recall the identity:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
We need to express both terms as squares.
2. Identify the Squares:
- The first term, [tex]\(16y^4\)[/tex], is a perfect square. It can be written as [tex]\((4y^2)^2\)[/tex].
- The second term, [tex]\(625x^4\)[/tex], is also a perfect square. It can be written as [tex]\((25x^2)^2\)[/tex].
So our expression becomes:
[tex]\[
(4y^2)^2 - (25x^2)^2
\][/tex]
3. Apply the Difference of Squares Formula:
Using the identity for the difference of squares:
[tex]\[
(4y^2)^2 - (25x^2)^2 = (4y^2 - 25x^2)(4y^2 + 25x^2)
\][/tex]
4. Further Factor the Resulting Expression:
Notice that [tex]\(4y^2 - 25x^2\)[/tex] is also a difference of squares:
- The first term, [tex]\(4y^2\)[/tex], is [tex]\((2y)^2\)[/tex].
- The second term, [tex]\(25x^2\)[/tex], is [tex]\((5x)^2\)[/tex].
Thus, applying the difference of squares identity again:
[tex]\[
4y^2 - 25x^2 = (2y - 5x)(2y + 5x)
\][/tex]
5. Write the Complete Factorization:
Combining everything, the full factorization of the original polynomial [tex]\(16y^4 - 625x^4\)[/tex] is:
[tex]\[
16y^4 - 625x^4 = (2y - 5x)(2y + 5x)(4y^2 + 25x^2)
\][/tex]
So, the polynomial [tex]\(16y^4 - 625x^4\)[/tex] is factored into [tex]\((2y - 5x)(2y + 5x)(4y^2 + 25x^2)\)[/tex].
1. Recognize it as a Difference of Squares:
The expression [tex]\(16y^4 - 625x^4\)[/tex] is in the form of a difference of squares. Recall the identity:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
We need to express both terms as squares.
2. Identify the Squares:
- The first term, [tex]\(16y^4\)[/tex], is a perfect square. It can be written as [tex]\((4y^2)^2\)[/tex].
- The second term, [tex]\(625x^4\)[/tex], is also a perfect square. It can be written as [tex]\((25x^2)^2\)[/tex].
So our expression becomes:
[tex]\[
(4y^2)^2 - (25x^2)^2
\][/tex]
3. Apply the Difference of Squares Formula:
Using the identity for the difference of squares:
[tex]\[
(4y^2)^2 - (25x^2)^2 = (4y^2 - 25x^2)(4y^2 + 25x^2)
\][/tex]
4. Further Factor the Resulting Expression:
Notice that [tex]\(4y^2 - 25x^2\)[/tex] is also a difference of squares:
- The first term, [tex]\(4y^2\)[/tex], is [tex]\((2y)^2\)[/tex].
- The second term, [tex]\(25x^2\)[/tex], is [tex]\((5x)^2\)[/tex].
Thus, applying the difference of squares identity again:
[tex]\[
4y^2 - 25x^2 = (2y - 5x)(2y + 5x)
\][/tex]
5. Write the Complete Factorization:
Combining everything, the full factorization of the original polynomial [tex]\(16y^4 - 625x^4\)[/tex] is:
[tex]\[
16y^4 - 625x^4 = (2y - 5x)(2y + 5x)(4y^2 + 25x^2)
\][/tex]
So, the polynomial [tex]\(16y^4 - 625x^4\)[/tex] is factored into [tex]\((2y - 5x)(2y + 5x)(4y^2 + 25x^2)\)[/tex].
Thank you for reading the article Factor the polynomial expression tex 16y 4 625x 4 tex tex 16y 4 625x 4 tex Choose from the following expressions tex 4y 2 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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