Factor p(x)=x^4-x^2-2

Problem

p(x)=x^4−x^2−2

Solution

1. Identify the expression as a trinomial in quadratic form by letting u=x^2

2. Substitute u into the expression to rewrite it as a quadratic trinomial.

p(u)=u^2−u−2

3. Factor the quadratic trinomial by finding two numbers that multiply to −2 and add to −1

p(u)=(u−2)*(u+1)

4. Substitute back x^2 for u to return to the original variable.

p(x)=(x^2−2)*(x^2+1)

5. Factor further if possible. The term (x^2−2) can be factored over the real numbers as a difference of squares.

x^2−2=(x−√(,2))*(x+√(,2))

Final Answer

x^4−x^2−2=(x−√(,2))*(x+√(,2))*(x^2+1)

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