Q. 1TB - Calculus

Question

Factoring formulas: Pull out a linear factor from each of the

following expressions.

$$\begin{gathered} z^2 - 100 \\ z^3 - 27 \\ z^6 - 64 \end{gathered}$$

Step-by-Step Solution

Answer

Linear factor from each of the expression

$$\begin{gathered} (z+10)(z-10) \\ (z-3) \\ (z+2)(z-2) \end{gathered}$$

Step 1: Given Information

Pull out a linear factor from each of the following expressions.

Part (1) : Step 1

$$\begin{gathered} z^2-100 \\ {z}^2-{10}^2 \\ (z+10)(z-10) \\ Linear factors (z+10)(z-10) \end{gathered}$$

Part(2) Step 1:

apply difference of cubes formula 

$$\begin{gathered} z^3-27 \\ {z}^2-3^3 \\ {x}^3-y^3=\left(x-y\right)\left(x^2+xy+y^2\right) \\ {z}^2-3^3 \\ \left(z-3\right)\left(z^2+3z+9\right) \\ Linear factor is (z-3) \end{gathered}$$

Part(3) Step 1:

$$\begin{gathered} z^6-64 \\ {\left(z^2\right)}^3-4^3 \\ \left(z^2-4\right)\left(z^4+4z^2+4^2\right) \\ \left(z+2\right)\left(z-2\right)\left(z^4+4z^2+4^2\right) \\ linear factor \\ \left(z+2\right)\left(z-2\right) \end{gathered}$$