find the factors of the following: (3x+ypower3z)
Answers
Answer:
Given: p(x,y)=405x^{6}+45y^{6}-270x^{3}y^{3}p(x,y)=405x
6
+45y
6
−270x
3
y
3
To prove: (\sqrt[3]{3}x-y)(
3
3
x−y) a factor of p(x,y)p(x,y)
Proof:
Now, 405x^{6}+45y^{6}-270x^{3}y^{3}405x
6
+45y
6
−270x
3
y
3
=45\:(9x^{6}+y^{6}-6x^{3}y^{3})=45(9x
6
+y
6
−6x
3
y
3
)
=45\:[(3x^{3})^{2}+(y^{3})^{2}-2\times 3x^{3}\times y^{3}=45[(3x
3
)
2
+(y
3
)
2
−2×3x
3
×y
3
=45\:(3x^{3}-y^{3})^{2}=45(3x
3
−y
3
)
2
=45\:\{(\sqrt[3]{3}x)^{3}-(y)^{3}\}^{2}=45{(
3
3
x)
3
−(y)
3
}
2
=45\:[(\sqrt[3]{3}x-y)\:\{(\sqrt[3]{3}x)^{2}+\sqrt[3]{3}xy+y^{2}\}]^{2}=45[(
3
3
x−y){(
3
3
x)
2
+
3
3
xy+y
2
}]
2
\longrightarrow⟶ This shows that (\sqrt[3]{3}x-y)(
3
3
x−y) is a factor of p(x,y)p(x,y) .
Hence proved.
Formula used:
a^{2}-2ab+b^{2}=(a-b)^{2}a
2
−2ab+b
2
=(a−b)
2
a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})a
3
−b
3
=(a−b)(a
2
+ab+b
2
)