(i) (6)/(5)x^(2)-(4)/(5)x^(3)+(5)/(6)+(3)/(2)x From (x^(3))/(3)-(5)/(2)x^(2)+(3)/(5)x+(1)/(4)

Suzhoumeite

3 min read

·

Jun 03, 2024

93

Share

Simplifying Algebraic Expressions

In this article, we will simplify the following algebraic expression:

$\frac{6}{5}x^2 - \frac{4}{5}x^3 + \frac{5}{6} + \frac{3}{2}x$

divided by

$\frac{x^3}{3} - \frac{5}{2}x^2 + \frac{3}{5}x + \frac{1}{4}$

Step 1: Simplify the numerator

The numerator is:

$\frac{6}{5}x^2 - \frac{4}{5}x^3 + \frac{5}{6} + \frac{3}{2}x$

To simplify the numerator, we can start by combining like terms:

$-\frac{4}{5}x^3 + \frac{6}{5}x^2 + \frac{3}{2}x + \frac{5}{6}$

Step 2: Simplify the denominator

The denominator is:

$\frac{x^3}{3} - \frac{5}{2}x^2 + \frac{3}{5}x + \frac{1}{4}$

To simplify the denominator, we can start by combining like terms:

$\frac{x^3}{3} - \frac{5}{2}x^2 + \frac{3}{5}x + \frac{1}{4}$

Step 3: Divide the numerator by the denominator

Now, we can divide the numerator by the denominator:

$\frac{-\frac{4}{5}x^3 + \frac{6}{5}x^2 + \frac{3}{2}x + \frac{5}{6}}{\frac{x^3}{3} - \frac{5}{2}x^2 + \frac{3}{5}x + \frac{1}{4}}$

To simplify this expression, we can start by multiplying both the numerator and the denominator by the least common multiple of the denominators, which is 60:

$\frac{60(-\frac{4}{5}x^3 + \frac{6}{5}x^2 + \frac{3}{2}x + \frac{5}{6})}{60(\frac{x^3}{3} - \frac{5}{2}x^2 + \frac{3}{5}x + \frac{1}{4})}$

Simplifying further, we get:

$\frac{-48x^3 + 72x^2 + 90x + 50}{20x^3 - 150x^2 + 36x + 15}$

And that's the simplified expression!