How do you factor the expression x cubed minus 5x plus 4?

Answer 1

By inspection, the sum of the coefficients is 1 - 5 + 4 = 0Therefore x = 1 is a root. That is (x - 1) is a factor.

Then suppose the other factor is ax^2 + bx + c

SInce the coefficient of x^3 is 1, then a = 1 so that the second factor is x^2 + bx + c

Therefore (x - 1)*(x^2 + bx + c) = x^3 - 5x + 4

x^3 - x^2 + bx^2 - bx + cx - c = x^3 - 5x + 4

Comparing coefficient of x^2 gives -1 + b = 0 so that b = 1

Comparing coefficient of x gives -b + c = -5 so that c = b - 5 = -4

Comparing constant terms gives -c = 4 or c = -4 which confirms above calculations.

Thus, the second factor is (x^2 + x - 4) which does not have rational factors.

Answer 2

(x - 1)(x^2 + x - 4)