Factorize:$ \frac{8}{27} x^{3}+1+\frac{4}{3} x^{2}+2 x $
Given:
\( \frac{8}{27} x^{3}+1+\frac{4}{3} x^{2}+2 x \)
To do:
We have to factorize the given expression.
Solution:
We know that,
$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$
$(a - b)^3 = a^3 - b^3 - 3ab(a - b)$
Therefore,
$\frac{8}{27} x^{3}+1+\frac{4}{3} x^{2}+2 x =(\frac{2}{3} x)^{3}+(1)^{3}+3 \times (\frac{2}{3} x)^{2} \times 1+3 \times \frac{2}{3} x \times(1)^{2}$
$=(\frac{2}{3} x+1)^{3}$
$=(\frac{2}{3} x+1)(\frac{2}{3} x+1)(\frac{2}{3} x+1)$
Hence, $\frac{8}{27} x^{3}+1+\frac{4}{3} x^{2}+2 x = (\frac{2}{3} x+1)(\frac{2}{3} x+1)(\frac{2}{3} x+1)$.
Related Articles
- Solve for $x$:$\frac{x-1}{x-2}+\frac{x-3}{x-4}=3\frac{1}{3}, x≠2, 4$
- If \( x+\frac{1}{x}=3 \), calculate \( x^{2}+\frac{1}{x^{2}}, x^{3}+\frac{1}{x^{3}} \) and \( x^{4}+\frac{1}{x^{4}} \).
- $\frac{x-1}{2}+\frac{2 x-1}{4}=\frac{x-1}{3}-\frac{2 x-1}{6}$.
- If \( x^{4}+\frac{1}{x^{4}}=194 \), find \( x^{3}+\frac{1}{x^{3}}, x^{2}+\frac{1}{x^{2}} \) and \( x+\frac{1}{x} \)
- Solve for $x$:\( 3 x-\frac{x-2}{3}=4-\frac{x-1}{4} \)
- Take away:\( \frac{6}{5} x^{2}-\frac{4}{5} x^{3}+\frac{5}{6}+\frac{3}{2} x \) from \( \frac{x^{3}}{3}-\frac{5}{2} x^{2}+\frac{3}{5} x+\frac{1}{4} \)
- Solve: $\frac{x-(7-8 x)}{9 x-(3+4 x)}=\frac{2}{3}$.
- Solve the following quadratic equation by factorization: $\frac{x-1}{x-2}+\frac{x-3}{x-4}=3\frac{1}{3}, x≠2, 4$
- Solve \( \frac{2 x+1}{3 x-2}=1\frac{1}{4} \).
- \Find $(x +y) \div (x - y)$. if,(i) \( x=\frac{2}{3}, y=\frac{3}{2} \)(ii) \( x=\frac{2}{5}, y=\frac{1}{2} \)(iii) \( x=\frac{5}{4}, y=\frac{-1}{3} \)(iv) \( x=\frac{2}{7}, y=\frac{4}{3} \)(v) \( x=\frac{1}{4}, y=\frac{3}{2} \)
- Solve the following:\( \frac{3}{4}(7 x-1)-\left(2 x-\frac{1-x}{2}\right)=x+\frac{3}{2} \)
- Solve the following pairs of equations by reducing them to a pair of linear equations:(i) \( \frac{1}{2 x}+\frac{1}{3 y}=2 \)\( \frac{1}{3 x}+\frac{1}{2 y}=\frac{13}{6} \)(ii) \( \frac{2}{\sqrt{x}}+\frac{3}{\sqrt{y}}=2 \)\( \frac{4}{\sqrt{x}}-\frac{9}{\sqrt{y}}=-1 \)(iii) \( \frac{4}{x}+3 y=14 \)\( \frac{3}{x}-4 y=23 \)(iv) \( \frac{5}{x-1}+\frac{1}{y-2}=2 \)\( \frac{6}{x-1}-\frac{3}{y-2}=1 \)(v) \( \frac{7 x-2 y}{x y}=5 \)\( \frac{8 x+7 y}{x y}=15 \),b>(vi) \( 6 x+3 y=6 x y \)\( 2 x+4 y=5 x y \)4(vii) \( \frac{10}{x+y}+\frac{2}{x-y}=4 \)\( \frac{15}{x+y}-\frac{5}{x-y}=-2 \)(viii) \( \frac{1}{3 x+y}+\frac{1}{3 x-y}=\frac{3}{4} \)\( \frac{1}{2(3 x+y)}-\frac{1}{2(3 x-y)}=\frac{-1}{8} \).
- Solve for x:$\frac{1}{( x-1)( x-2)} +\frac{1}{( x-2)( x-3)} =\frac{2}{3} \ ,\ x\neq 1,2,3$
- Solve for $x$:$\frac{1}{x}+\frac{2}{2x-3}=\frac{1}{x-2}, x≠0, \frac{3}{2}, 2$
- If $x - \frac{1}{x} = 3$, find the values of $x^2 + \frac{1}{x^2}$ and $x^4 + \frac{1}{x^4}$.
Kickstart Your Career
Get certified by completing the course
Get Started
Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
Physics
Chemistry
Biology
Mathematics
English
Economics
Psychology
Social Studies
Fashion Studies
Legal Studies