Chapter A.2, Problem 53E

Solution

To factor the sum or difference of two cubes $y^3-8$ .

  $\left(y-2\right)\left(y^2+2y+4\right)$

Given:

The expression, $y^3-8$

Concept Used:

• Like terms are the terms that contain the same variable with similar exponents on it. Constants are like terms as well.
• In any expression, terms are separated by plus or minus signs.
• Coefficients are the constants in front of the variable terms.
• $a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)$

Calculation:

In order to find the factor the of two cubes first write -8 in exponential term then solving the equation by using the difference of cubes formula as shown below

  $8=2^3$

  $y^3-2^3$

  $\begin{array}{l}{a}^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)\\ y^3-2^3=\left(y-2\right)\left(y^2+2\cdot y+2^2\right)\\ =\left(y-2\right)\left(y^2+2y+4\right)\end{array}$

The original expression now become

  $\left(y-2\right)\left(y^2+2y+4\right)$