Question

which expression shows the complete factored form of $x^4 - 3x^3 - x^2 + 3x$?
a $x(x - 1)(x + 1)(x + 3)$
b $x(x^2 - 1)(x + 3)$
c $x(x - 1)(x + 1)(x - 3)$
d $x(x^2 + 1)(x - 3)$

Explanation:

Step1: Factor out common terms

First, factor out \( x \) from the polynomial \( x^4 - 3x^3 - x^2 + 3x \). We get \( x(x^3 - 3x^2 - x + 3) \).

Step2: Group terms in the cubic

Group the terms in the cubic expression \( x^3 - 3x^2 - x + 3 \) as \( (x^3 - 3x^2) + (-x + 3) \).

Step3: Factor each group

Factor out \( x^2 \) from the first group: \( x^2(x - 3) \), and factor out \( -1 \) from the second group: \( -1(x - 3) \). Now the expression becomes \( x^2(x - 3) - 1(x - 3) \).

Step4: Factor out common binomial

Factor out \( (x - 3) \) from the above expression: \( (x^2 - 1)(x - 3) \).

Step5: Factor difference of squares

Factor \( x^2 - 1 \) as a difference of squares: \( (x - 1)(x + 1) \).

Step6: Combine all factors

Putting it all together, the original polynomial factors to \( x(x - 1)(x + 1)(x - 3) \).

Answer:

C. \( x(x - 1)(x + 1)(x - 3) \)