Solve (a + 1)/(a ^ 2 + a + 1) + (a - 1)/(a ^ 2 - a + 1) + 2/(a ^ 4 + a ^ 2 + 1) | 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

$$\frac{a+1}{a^{2}+a+1}+\frac{a-1}{a^{2}-a+1}+\frac{2}{a^{4}+a^{2}+1}$$

Evaluate

$\frac{2\left(a+1\right)}{a^{2}+a+1}$

Short Solution Steps

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

$$\frac{\left(a+1\right)\left(a^{2}-a+1\right)}{\left(a^{2}+a+1\right)\left(a^{2}-a+1\right)}+\frac{\left(a-1\right)\left(a^{2}+a+1\right)}{\left(a^{2}+a+1\right)\left(a^{2}-a+1\right)}+\frac{2}{a^{4}+a^{2}+1}$$

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

$$\frac{\left(a+1\right)\left(a^{2}-a+1\right)+\left(a-1\right)\left(a^{2}+a+1\right)}{\left(a^{2}+a+1\right)\left(a^{2}-a+1\right)}+\frac{2}{a^{4}+a^{2}+1}$$

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

$$\frac{a^{3}-a^{2}+a+a^{2}-a+1+a^{3}+a^{2}+a-a^{2}-a-1}{\left(a^{2}+a+1\right)\left(a^{2}-a+1\right)}+\frac{2}{a^{4}+a^{2}+1}$$

Combine like terms in $a^{3}-a^{2}+a+a^{2}-a+1+a^{3}+a^{2}+a-a^{2}-a-1$.

$$\frac{2a^{3}}{\left(a^{2}+a+1\right)\left(a^{2}-a+1\right)}+\frac{2}{a^{4}+a^{2}+1}$$

Factor $a^{4}+a^{2}+1$.

$$\frac{2a^{3}}{\left(a^{2}+a+1\right)\left(a^{2}-a+1\right)}+\frac{2}{\left(a^{2}+a+1\right)\left(a^{2}-a+1\right)}$$

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

$$\frac{2a^{3}+2}{\left(a^{2}+a+1\right)\left(a^{2}-a+1\right)}$$

Factor the expressions that are not already factored in $\frac{2a^{3}+2}{\left(a^{2}+a+1\right)\left(a^{2}-a+1\right)}$.

$$\frac{2\left(a+1\right)\left(a^{2}-a+1\right)}{\left(a^{2}+a+1\right)\left(a^{2}-a+1\right)}$$

Cancel out $a^{2}-a+1$ in both numerator and denominator.

$$\frac{2\left(a+1\right)}{a^{2}+a+1}$$

Use the distributive property to multiply $2$ by $a+1$.

$$\frac{2a+2}{a^{2}+a+1}$$

Expand

$\frac{2\left(a+1\right)}{a^{2}+a+1}$

Short Solution Steps

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

$$\frac{\left(a+1\right)\left(a^{2}-a+1\right)}{\left(a^{2}+a+1\right)\left(a^{2}-a+1\right)}+\frac{\left(a-1\right)\left(a^{2}+a+1\right)}{\left(a^{2}+a+1\right)\left(a^{2}-a+1\right)}+\frac{2}{a^{4}+a^{2}+1}$$

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

$$\frac{\left(a+1\right)\left(a^{2}-a+1\right)+\left(a-1\right)\left(a^{2}+a+1\right)}{\left(a^{2}+a+1\right)\left(a^{2}-a+1\right)}+\frac{2}{a^{4}+a^{2}+1}$$

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

$$\frac{a^{3}-a^{2}+a+a^{2}-a+1+a^{3}+a^{2}+a-a^{2}-a-1}{\left(a^{2}+a+1\right)\left(a^{2}-a+1\right)}+\frac{2}{a^{4}+a^{2}+1}$$

Combine like terms in $a^{3}-a^{2}+a+a^{2}-a+1+a^{3}+a^{2}+a-a^{2}-a-1$.

$$\frac{2a^{3}}{\left(a^{2}+a+1\right)\left(a^{2}-a+1\right)}+\frac{2}{a^{4}+a^{2}+1}$$

Factor $a^{4}+a^{2}+1$.

$$\frac{2a^{3}}{\left(a^{2}+a+1\right)\left(a^{2}-a+1\right)}+\frac{2}{\left(a^{2}+a+1\right)\left(a^{2}-a+1\right)}$$

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

$$\frac{2a^{3}+2}{\left(a^{2}+a+1\right)\left(a^{2}-a+1\right)}$$

Factor the expressions that are not already factored in $\frac{2a^{3}+2}{\left(a^{2}+a+1\right)\left(a^{2}-a+1\right)}$.

$$\frac{2\left(a+1\right)\left(a^{2}-a+1\right)}{\left(a^{2}+a+1\right)\left(a^{2}-a+1\right)}$$

Cancel out $a^{2}-a+1$ in both numerator and denominator.

$$\frac{2\left(a+1\right)}{a^{2}+a+1}$$

Use the distributive property to multiply $2$ by $a+1$.

$$\frac{2a+2}{a^{2}+a+1}$$