Question

Factor each trinomial, if possible. If the polynomial cannot be factored, write prime.

${\mathit{x}}^{\mathbf{4}}\mathbf{-}\mathbf{1}$

Short answer

The factor of the polynomial is:

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

Step-by-step solution

Step 1. State the concept for factorization of algebraic expressions.

I. The product of two factors with like signs is positive, and the product of two factors with unlike signs is negative.

II. If x is a variable and m, n are positive integers, then$x^{m+n}=\left(x^m\times x^n\right)$

Step 2. State the methods to factorize.

Factoring an algebraic expression means writing the expression as a product of factors.

To verify whether the factors are correct or not, multiply them and check if the result is the original algebraic expression.

Algebraic expressions can be factorized using the common factor method, regrouping like terms together, and also by using algebraic identities.

Step 3. Factorize the expression.

Factorize the given polynomial by using the formula: $a^2-b^2=\left(a-b\right)\left(a+b\right)$.

$\begin{array}{c}{x}^4-1=\left({\left(x^2\right)}^2-1^2\right)\\ =\left(x^2-1\right)\left(x^2+1\right)\\ =\left(x^2-1^2\right)\left(x^2+1\right)\\ =\left(x-1\right)\left(x+1\right)\left(x^2+1\right)\end{array}$

Therefore, the factor is $\left(x-1\right)\left(x+1\right)\left(x^2+1\right)$.