Question

1. $x^{3}+y^{3}=$?

a. $(x - y)(x^{2}+xy - y^{2})$
b. $sqrt{9x^{2}+9y^{2}}$
c. $sqrt3{x^{6}+y^{6}}$
d. $(x + y)(x^{2}-xy + y^{2})$
e. $(x - y)(x + y)^{2}$

Explanation:

Step1: Recall the formula for sum of cubes

The formula for the sum of two cubes is \(a^3 + b^3=(a + b)(a^2 - ab + b^2)\). Here, \(a = x\) and \(b = y\), so \(x^3 + y^3=(x + y)(x^2 - xy + y^2)\).

Step2: Analyze option A

For option A, \((x - y)(x^2 + xy - y^2)\) expands to \(x^3 + x^2y - xy^2 - x^2y - xy^2 + y^3=x^3 - 2xy^2 + y^3\), which is not \(x^3 + y^3\).

Step3: Analyze option B

\(\sqrt{9x^2 + 9y^2}=\sqrt{9(x^2 + y^2)} = 3\sqrt{x^2 + y^2}\), which is not related to \(x^3 + y^3\).

Step4: Analyze option C

\(\sqrt[3]{x^6 + y^6}=\sqrt[3]{(x^2)^3+(y^2)^3}=(x^2 + y^2)\) (by sum of cubes formula for \(a=x^2\), \(b = y^2\)), which is not \(x^3 + y^3\).

Step5: Analyze option D

Using the sum of cubes formula as in Step 1, \((x + y)(x^2 - xy + y^2)=x^3 - x^2y + xy^2 + x^2y - xy^2 + y^3=x^3 + y^3\), which matches.

Step6: Analyze option E

\((x - y)(x + y)^2=(x - y)(x^2 + 2xy + y^2)=x^3 + 2x^2y + xy^2 - x^2y - 2xy^2 - y^3=x^3 + x^2y - xy^2 - y^3\), which is not \(x^3 + y^3\).

Answer:

D. \((x + y)(x^2 - xy + y^2)\)