Solve f(z) = 1/((z ^ 2 + 1)(z ^ 2 + 2)) | Equatix - Math Solver

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Example

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

a⋅(b-2)=3b

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

a⋅(b-2)=3b

Question

$$f(z)=\frac{1}{(z^{2}+1)(z^{2}+2)}$$

Solve for f

$f=\frac{1}{z\left(z^{4}+3z^{2}+2\right)}$
$z\neq 0$

Steps for Solving Linear Equation

Multiply both sides of the equation by $\left(z^{2}+1\right)\left(z^{2}+2\right)$.

$$fz\left(z^{2}+1\right)\left(z^{2}+2\right)=1$$

Use the distributive property to multiply $fz$ by $z^{2}+1$.

$$\left(fz^{3}+fz\right)\left(z^{2}+2\right)=1$$

Use the distributive property to multiply $fz^{3}+fz$ by $z^{2}+2$ and combine like terms.

$$fz^{5}+3fz^{3}+2fz=1$$

Combine all terms containing $f$.

$$\left(z^{5}+3z^{3}+2z\right)f=1$$

Divide both sides by $z^{5}+3z^{3}+2z$.

$$\frac{\left(z^{5}+3z^{3}+2z\right)f}{z^{5}+3z^{3}+2z}=\frac{1}{z^{5}+3z^{3}+2z}$$

Dividing by $z^{5}+3z^{3}+2z$ undoes the multiplication by $z^{5}+3z^{3}+2z$.

$$f=\frac{1}{z^{5}+3z^{3}+2z}$$

Divide $1$ by $z^{5}+3z^{3}+2z$.

$$f=\frac{1}{z\left(z^{4}+3z^{2}+2\right)}$$