Question

write the expansion (2/3x-5/3z)³. ​

Answer 1

Question:

Expand the Following : $\Big (\dfrac{2}{3} x-\dfrac{5}{3} z \Big )^3$

$\begin {gathered} \end {gathered}$

Identity Used :-

$\boxed {\begin {aligned} \mathtt {(a - b)^3} & = \mathtt {a^3 - b^3 - 3ab\ (a - b)} \\\\& = \mathtt {a^3 - b^3 - 3a^2b + 3ab^2} \end {aligned}}$

Where, a = $\bf \dfrac{2}{3}x$ ; b = $\bf \dfrac{5}{3}z$

$\begin {gathered} \end {gathered}$

Solution :-

We have :

$\begin {aligned} \bold {\Big (\dfrac{2}{3}x - \dfrac{5}{3}z \Big )^3} & = \Big (\dfrac{2}{3} x \Big )^3 - \Big (\dfrac{5}{3} z \Big )^3 - 3\times \dfrac{2}{3} x \times \dfrac{5}{3} z\ \Big [\dfrac{2}{3} x - \dfrac{5}{3} z \Big ]\\\\& \Rightarrow \dfrac{8}{27} x^3 - \dfrac{125}{27} z^3 - \dfrac{30}{9} xz\ \Big [\dfrac{2}{3} x - \dfrac{5}{3} z \Big ]\\\\& = \bold {\dfrac{8}{27} x^3 - \dfrac{125}{27} z^3 - \dfrac{60}{27} x^2z + \dfrac{150}{27} xz^2} \end {aligned}$

$\begin {gathered} \end {gathered}$

∴ The Expanded form of $\bf \Big (\dfrac{2}{3} x - \dfrac{5}{3} z \Big )^3$ is $\bf \dfrac{8}{27} x^3 - \dfrac{125}{27} z^3 - \dfrac{60}{27} x^2z + \dfrac{150}{27} xz^2$

$\begin {gathered} \end {gathered}$

Additional Information :-

Some more Expansions :-

i. $\mathtt {(a + b)^2 = a^2 + 2ab + b^2}$

ii. $\mathtt{(a - b)^2 = a^2 - 2ab + b^2}$

iii. $\mathtt{(a + b + c)^2 = a^2 + b^2 + c^2+ 2ab + 2bc + 2ca}$

iv. $\mathtt{(a + b)^3 = a^3+b^3+3ab\ (a+b)}$

v. $\mathtt{(a^2 - b^2) = (a - b)(a + b)}$

$\begin {gathered} \end {gathered}$

Factorization :- To Express a given Polynomial as the Product of Polynomials, each of Degree less than that of the given Polynomial such that no such a Factor has a Factor of Lower Degree, is called Factorization . . .