Chapter A.2, Problem 71E

Solution

To factor by grouping $x^6-3x^4+x^2-3$ .

  $\left(x^4+1\right)\left(x^2-3\right)$

Given:

The expression, $x^6-3x^4+x^2-3$

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.

Calculation:

In order to find the factor by grouping, first group the first two terms and the last two terms and then factor out the greatest common factor from each group and then factor the polynomial by factoring out the greatest common factor, $x^2-3$ as shown below:

  $\begin{array}{l}{x}^6-3x^4+x^2-3\\ \left(x^6-3x^4\right)+\left(x^2-3\right)\\ x^4\left(x^2-3\right)+1\left(x^2-3\right)\\ \left(x^4+1\right)\left(x^2-3\right)\end{array}$

The original expression now become

  $\left(x^4+1\right)\left(x^2-3\right)$