Answer :
Let's evaluate each option to determine which one is a true polynomial identity:
a) [tex]\((x^3 + y^2)(x^3 + y^2) = x^6 + y^4\)[/tex]
When expanding the left side, [tex]\((x^3 + y^2)(x^3 + y^2)\)[/tex], it follows:
- First, distribute the first term: [tex]\(x^3 \cdot x^3 = x^6\)[/tex].
- Then distribute [tex]\(x^3 \cdot y^2 = x^3y^2\)[/tex].
- Next, distribute [tex]\(y^2 \cdot x^3 = x^3y^2\)[/tex].
- Finally, distribute [tex]\(y^2 \cdot y^2 = y^4\)[/tex].
Combining these results, the expression expands to:
[tex]\[ x^6 + 2x^3y^2 + y^4 \][/tex]
This does not match [tex]\(x^6 + y^4\)[/tex], so option (a) is false.
b) [tex]\((x^3 - y^2)(x^3 + y^2) + 2y^4 = x^6 - y^4\)[/tex]
First, expand [tex]\((x^3 - y^2)(x^3 + y^2)\)[/tex]:
- This is a difference of squares: [tex]\(x^3(x^3) - y^2(y^2)\)[/tex].
- So, it becomes [tex]\(x^6 - y^4\)[/tex].
Now, add [tex]\(2y^4\)[/tex]:
[tex]\[ x^6 - y^4 + 2y^4 = x^6 + y^4 \][/tex]
This does not match [tex]\(x^6 - y^4\)[/tex], so option (b) is false.
c) [tex]\((x^3 - y^2)(x^3 + y^2) = x^6 - 2y^4\)[/tex]
Again, using the difference of squares:
- [tex]\(x^3(x^3) - y^2(y^2) = x^6 - y^4\)[/tex].
This does not match [tex]\(x^6 - 2y^4\)[/tex], so option (c) is also false.
d) [tex]\((x^3 - y^2)(x^3 + y^2) + 2y^4 = x^6 + y^4\)[/tex]
Using the same expansion as before:
- [tex]\((x^3 - y^2)(x^3 + y^2) = x^6 - y^4\)[/tex].
Now, add [tex]\(2y^4\)[/tex]:
[tex]\[ x^6 - y^4 + 2y^4 = x^6 + y^4 \][/tex]
This matches exactly, so option (d) is true.
Therefore, the true polynomial identity is option (d):
[tex]\((x^3 - y^2)(x^3 + y^2) + 2y^4 = x^6 + y^4\)[/tex].
a) [tex]\((x^3 + y^2)(x^3 + y^2) = x^6 + y^4\)[/tex]
When expanding the left side, [tex]\((x^3 + y^2)(x^3 + y^2)\)[/tex], it follows:
- First, distribute the first term: [tex]\(x^3 \cdot x^3 = x^6\)[/tex].
- Then distribute [tex]\(x^3 \cdot y^2 = x^3y^2\)[/tex].
- Next, distribute [tex]\(y^2 \cdot x^3 = x^3y^2\)[/tex].
- Finally, distribute [tex]\(y^2 \cdot y^2 = y^4\)[/tex].
Combining these results, the expression expands to:
[tex]\[ x^6 + 2x^3y^2 + y^4 \][/tex]
This does not match [tex]\(x^6 + y^4\)[/tex], so option (a) is false.
b) [tex]\((x^3 - y^2)(x^3 + y^2) + 2y^4 = x^6 - y^4\)[/tex]
First, expand [tex]\((x^3 - y^2)(x^3 + y^2)\)[/tex]:
- This is a difference of squares: [tex]\(x^3(x^3) - y^2(y^2)\)[/tex].
- So, it becomes [tex]\(x^6 - y^4\)[/tex].
Now, add [tex]\(2y^4\)[/tex]:
[tex]\[ x^6 - y^4 + 2y^4 = x^6 + y^4 \][/tex]
This does not match [tex]\(x^6 - y^4\)[/tex], so option (b) is false.
c) [tex]\((x^3 - y^2)(x^3 + y^2) = x^6 - 2y^4\)[/tex]
Again, using the difference of squares:
- [tex]\(x^3(x^3) - y^2(y^2) = x^6 - y^4\)[/tex].
This does not match [tex]\(x^6 - 2y^4\)[/tex], so option (c) is also false.
d) [tex]\((x^3 - y^2)(x^3 + y^2) + 2y^4 = x^6 + y^4\)[/tex]
Using the same expansion as before:
- [tex]\((x^3 - y^2)(x^3 + y^2) = x^6 - y^4\)[/tex].
Now, add [tex]\(2y^4\)[/tex]:
[tex]\[ x^6 - y^4 + 2y^4 = x^6 + y^4 \][/tex]
This matches exactly, so option (d) is true.
Therefore, the true polynomial identity is option (d):
[tex]\((x^3 - y^2)(x^3 + y^2) + 2y^4 = x^6 + y^4\)[/tex].
