Løs 1/x^2-x-2+3/x^2-4-1/x^2+3x+2

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

Evaluer

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

Korte løsningstrinn

Faktoriser x^{2}-x-2. Faktoriser x^{2}-4.

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

Hvis du vil legge til eller trekke fra uttrykk, kan du utvide dem for å gjøre nevnerne like. Minste felles multiplum av \left(x-2\right)\left(x+1\right) og \left(x-2\right)\left(x+2\right) er \left(x-2\right)\left(x+1\right)\left(x+2\right). Multipliser \frac{1}{\left(x-2\right)\left(x+1\right)} ganger \frac{x+2}{x+2}. Multipliser \frac{3}{\left(x-2\right)\left(x+2\right)} ganger \frac{x+1}{x+1}.

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

Siden \frac{x+2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)} og \frac{3\left(x+1\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)} har samme nevner, kan du legge dem sammen ved å legge sammen tellerne.

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

Utfør multiplikasjonene i x+2+3\left(x+1\right).

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

Kombiner like ledd i x+2+3x+3.

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

Faktoriser x^{2}+3x+2.

$$\frac{4x+5}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}-\frac{1}{\left(x+1\right)\left(x+2\right)}$$

Hvis du vil legge til eller trekke fra uttrykk, kan du utvide dem for å gjøre nevnerne like. Minste felles multiplum av \left(x-2\right)\left(x+1\right)\left(x+2\right) og \left(x+1\right)\left(x+2\right) er \left(x-2\right)\left(x+1\right)\left(x+2\right). Multipliser \frac{1}{\left(x+1\right)\left(x+2\right)} ganger \frac{x-2}{x-2}.

$$\frac{4x+5}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}-\frac{x-2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}$$

Siden \frac{4x+5}{\left(x-2\right)\left(x+1\right)\left(x+2\right)} og \frac{x-2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)} har samme nevner, kan du subtrahere dem ved å subtrahere tellerne.

$$\frac{4x+5-\left(x-2\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}$$

Utfør multiplikasjonene i 4x+5-\left(x-2\right).

$$\frac{4x+5-x+2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}$$

Kombiner like ledd i 4x+5-x+2.

$$\frac{3x+7}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}$$

Utvid \left(x-2\right)\left(x+1\right)\left(x+2\right).

$$\frac{3x+7}{x^{3}+x^{2}-4x-4}$$

Differensier med hensyn til x

$$\frac{2\left(8-7x-12x^{2}-3x^{3}\right)}{\left(\left(x+1\right)\left(x^{2}-4\right)\right)^{2}}$$