Chapter 8, Problem 29PT

To determine

To factor the given polynomial

Answer to Problem 29PT

  $(x+4)(x+1)(x-1)$

Explanation of Solution

Given:

Polynomial: $x^3+4x^2-x-4$

Calculation:

Consider the given polynomial,

  $\Rightarrow x^3+4x^2-x-4$

Grouping the like terms,

  $\Rightarrow (x^3-x)+(4x^2-4)$

Find out common terms in the above expression.

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

Here, $(x^2-1)$ is a common term,

  $\begin{array}{l}\Rightarrow x(x^2-1)+4(x^2-1)\\ \Rightarrow (x+4)(x^2-1)\end{array}$

Further, $(x^2-1)$ can be factored as,

  $\begin{array}{l}\Rightarrow (x^2-1)\\ $ \Rightarrow ({(x)}^2-{(1)}^2)\left[\because (a^2-b^2)=(a+b)(a-b)\right]$\Rightarrow (x+1)(x-1)\end{array}$

Therefore, the polynomial can be factorised as,

  $\Rightarrow (x+4)(x+1)(x-1)$

Conclusion:

Therefore, the polynomial is factored as $(x+4)(x+1)(x-1).$