Solve z^3+5z+3z^2+1-4-2z^2

$$z ^ { 3 } + 5 z + 3 z ^ { 2 } + 1 - 4 - 2 z ^ { 2 }$$

Evaluate

$$z^{3}+z^{2}+5z-3$$

Solution Steps

Subtract 4 from 1 to get -3.

$$z^{3}+5z+3z^{2}-3-2z^{2}$$

Combine 3z^{2} and -2z^{2} to get z^{2}.

$$z^{3}+5z+z^{2}-3$$

Differentiate w.r.t. z

$$3z^{2}+2z+5$$

Steps Using Derivative Rule for Sum

Subtract 4 from 1 to get -3.

$$\frac{\mathrm{d}}{\mathrm{d}z}(z^{3}+5z+3z^{2}-3-2z^{2})$$

Combine 3z^{2} and -2z^{2} to get z^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}z}(z^{3}+5z+z^{2}-3)$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$3z^{3-1}+5z^{1-1}+2z^{2-1}$$

Subtract 1 from 3.

$$3z^{2}+5z^{1-1}+2z^{2-1}$$

Subtract 1 from 1.

$$3z^{2}+5z^{0}+2z^{2-1}$$

Subtract 1 from 2.

$$3z^{2}+5z^{0}+2z^{1}$$

For any term t, t^{1}=t.

$$3z^{2}+5z^{0}+2z$$

For any term t except 0, t^{0}=1.

$$3z^{2}+5\times 1+2z$$

For any term t, t\times 1=t and 1t=t.

$$3z^{2}+5+2z$$