Solve Date Page . 2/(x ^ 2 + 3x + 2) + (5x)/(x ^ 2 - x - 6) + (x + 2)/(x ^ 2 - 2x - 3) | Equatix - Math Solver

Fractions & Decimals

Ratios & Proportions

Basic Number Theory

Integers & Absolute Values

Factors & Multiples

Prime Factorization

Basic Equations & Inequalities

Coordinate Plane & Graphing

Linear Equations & Inequalities

Quadratic Equations

Polynomials & Factoring

Functions & Graphs

Systems of Equations

Trigonometric Functions

Trigonometric Identities & Equations

Right Triangle Trigonometry

Graphing Trigonometric Functions

Limits & Continuity

Derivatives & Applications

Integrals & Applications

Series & Sequences

Multivariable Calculus

Functions & Graphs

Polynomial & Rational Functions

Exponential & Logarithmic Functions

Sequences & Series

Descriptive Statistics

Normal Distribution

Hypothesis Testing

Regression Analysis

Linear Programming

Sets & Counting

Eigenvalues & Eigenvectors

Linear Transformations

Atomic Structure

Thermochemistry

Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

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

Evaluate

$\frac{x^{4}+20x^{3}+57x^{2}+41x+7}{3\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}$

Short Solution Steps

Combine $x$ and $2x$ to get $3x$.

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

Factor $x^{2}+3x+2$.

$$\frac{1}{\left(x+1\right)\left(x+2\right)}+\frac{5x}{x^{2}+x+1}+\frac{x+2}{3x+3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x+1\right)\left(x+2\right)$ and $x^{2}+x+1$ is $\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)$. Multiply $\frac{1}{\left(x+1\right)\left(x+2\right)}$ times $\frac{x^{2}+x+1}{x^{2}+x+1}$. Multiply $\frac{5x}{x^{2}+x+1}$ times $\frac{\left(x+1\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}$.

$$\frac{x^{2}+x+1}{\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}+\frac{5x\left(x+1\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}+\frac{x+2}{3x+3}$$

Since $\frac{x^{2}+x+1}{\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}$ and $\frac{5x\left(x+1\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{x^{2}+x+1+5x\left(x+1\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}+\frac{x+2}{3x+3}$$

Do the multiplications in $x^{2}+x+1+5x\left(x+1\right)\left(x+2\right)$.

$$\frac{x^{2}+x+1+5x^{3}+10x^{2}+5x^{2}+10x}{\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}+\frac{x+2}{3x+3}$$

Combine like terms in $x^{2}+x+1+5x^{3}+10x^{2}+5x^{2}+10x$.

$$\frac{16x^{2}+11x+1+5x^{3}}{\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}+\frac{x+2}{3x+3}$$

Factor $3x+3$.

$$\frac{16x^{2}+11x+1+5x^{3}}{\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}+\frac{x+2}{3\left(x+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)$ and $3\left(x+1\right)$ is $3\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)$. Multiply $\frac{16x^{2}+11x+1+5x^{3}}{\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}$ times $\frac{3}{3}$. Multiply $\frac{x+2}{3\left(x+1\right)}$ times $\frac{\left(x+2\right)\left(x^{2}+x+1\right)}{\left(x+2\right)\left(x^{2}+x+1\right)}$.

$$\frac{3\left(16x^{2}+11x+1+5x^{3}\right)}{3\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}+\frac{\left(x+2\right)\left(x+2\right)\left(x^{2}+x+1\right)}{3\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}$$

Since $\frac{3\left(16x^{2}+11x+1+5x^{3}\right)}{3\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}$ and $\frac{\left(x+2\right)\left(x+2\right)\left(x^{2}+x+1\right)}{3\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{3\left(16x^{2}+11x+1+5x^{3}\right)+\left(x+2\right)\left(x+2\right)\left(x^{2}+x+1\right)}{3\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}$$

Do the multiplications in $3\left(16x^{2}+11x+1+5x^{3}\right)+\left(x+2\right)\left(x+2\right)\left(x^{2}+x+1\right)$.

$$\frac{48x^{2}+33x+3+15x^{3}+x^{4}+3x^{3}+3x^{2}+2x+2x^{3}+6x^{2}+6x+4}{3\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}$$

Combine like terms in $48x^{2}+33x+3+15x^{3}+x^{4}+3x^{3}+3x^{2}+2x+2x^{3}+6x^{2}+6x+4$.

$$\frac{57x^{2}+41x+7+20x^{3}+x^{4}}{3\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}$$

Expand $3\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)$.

$$\frac{57x^{2}+41x+7+20x^{3}+x^{4}}{3x^{4}+12x^{3}+18x^{2}+15x+6}$$

Expand

$\frac{x^{4}+20x^{3}+57x^{2}+41x+7}{3\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}$

Short Solution Steps

Combine $x$ and $2x$ to get $3x$.

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

Factor $x^{2}+3x+2$.

$$\frac{1}{\left(x+1\right)\left(x+2\right)}+\frac{5x}{x^{2}+x+1}+\frac{x+2}{3x+3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x+1\right)\left(x+2\right)$ and $x^{2}+x+1$ is $\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)$. Multiply $\frac{1}{\left(x+1\right)\left(x+2\right)}$ times $\frac{x^{2}+x+1}{x^{2}+x+1}$. Multiply $\frac{5x}{x^{2}+x+1}$ times $\frac{\left(x+1\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}$.

$$\frac{x^{2}+x+1}{\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}+\frac{5x\left(x+1\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}+\frac{x+2}{3x+3}$$

Since $\frac{x^{2}+x+1}{\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}$ and $\frac{5x\left(x+1\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{x^{2}+x+1+5x\left(x+1\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}+\frac{x+2}{3x+3}$$

Do the multiplications in $x^{2}+x+1+5x\left(x+1\right)\left(x+2\right)$.

$$\frac{x^{2}+x+1+5x^{3}+10x^{2}+5x^{2}+10x}{\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}+\frac{x+2}{3x+3}$$

Combine like terms in $x^{2}+x+1+5x^{3}+10x^{2}+5x^{2}+10x$.

$$\frac{16x^{2}+11x+1+5x^{3}}{\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}+\frac{x+2}{3x+3}$$

Factor $3x+3$.

$$\frac{16x^{2}+11x+1+5x^{3}}{\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}+\frac{x+2}{3\left(x+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)$ and $3\left(x+1\right)$ is $3\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)$. Multiply $\frac{16x^{2}+11x+1+5x^{3}}{\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}$ times $\frac{3}{3}$. Multiply $\frac{x+2}{3\left(x+1\right)}$ times $\frac{\left(x+2\right)\left(x^{2}+x+1\right)}{\left(x+2\right)\left(x^{2}+x+1\right)}$.

$$\frac{3\left(16x^{2}+11x+1+5x^{3}\right)}{3\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}+\frac{\left(x+2\right)\left(x+2\right)\left(x^{2}+x+1\right)}{3\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}$$

Since $\frac{3\left(16x^{2}+11x+1+5x^{3}\right)}{3\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}$ and $\frac{\left(x+2\right)\left(x+2\right)\left(x^{2}+x+1\right)}{3\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{3\left(16x^{2}+11x+1+5x^{3}\right)+\left(x+2\right)\left(x+2\right)\left(x^{2}+x+1\right)}{3\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}$$

Do the multiplications in $3\left(16x^{2}+11x+1+5x^{3}\right)+\left(x+2\right)\left(x+2\right)\left(x^{2}+x+1\right)$.

$$\frac{48x^{2}+33x+3+15x^{3}+x^{4}+3x^{3}+3x^{2}+2x+2x^{3}+6x^{2}+6x+4}{3\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}$$

Combine like terms in $48x^{2}+33x+3+15x^{3}+x^{4}+3x^{3}+3x^{2}+2x+2x^{3}+6x^{2}+6x+4$.

$$\frac{57x^{2}+41x+7+20x^{3}+x^{4}}{3\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)}$$

Expand $3\left(x+1\right)\left(x+2\right)\left(x^{2}+x+1\right)$.

$$\frac{57x^{2}+41x+7+20x^{3}+x^{4}}{3x^{4}+12x^{3}+18x^{2}+15x+6}$$