user
How to factorise x**3-x+6 ?
alphonsio

The factorization of x3−x+6x^3 - x + 6 is :

x3−x+6=(x+2)(x2−2x+3)x^3 - x + 6 = (x + 2)(x^2 - 2x + 3)

Graph of the function y = x³-x+6

To factorize the polynomial x3−x+6x^3 - x + 6, we'll use a combination of techniques, including the Rational Root Theorem and polynomial division.

Step 1: Apply the Rational Root Theorem

The Rational Root Theorem states that any possible rational root of the polynomial f(x)=x3−x+6f(x) = x^3 - x + 6 must be a factor of the constant term (here, 6) divided by a factor of the leading coefficient (here, 1).

The factors of 6 are: ±1,±2,±3,±6\pm 1, \pm 2, \pm 3, \pm 6
The factors of 1 are: ±1\pm 1

So, the possible rational roots could be: ±1,±2,±3,±6\pm 1, \pm 2, \pm 3, \pm 6

Step 2: Test possible rational roots

We will test these values by substituting them into the polynomial f(x)=x3−x+6f(x) = x^3 - x + 6:

  • For x=1x = 1:
    13−1+6=1−1+6=6(not a root)1^3 - 1 + 6 = 1 - 1 + 6 = 6 \quad (\text{not a root})

  • For x=−1x = -1:
    (−1)3−(−1)+6=−1+1+6=6(not a root)(-1)^3 - (-1) + 6 = -1 + 1 + 6 = 6 \quad (\text{not a root})

  • For x=2x = 2:
    23−2+6=8−2+6=12(not a root)2^3 - 2 + 6 = 8 - 2 + 6 = 12 \quad (\text{not a root})

  • For x=−2x = -2:
    (−2)3−(−2)+6=−8+2+6=0(root found)(-2)^3 - (-2) + 6 = -8 + 2 + 6 = 0 \quad (\text{root found})

Since x=−2x = -2 is a root, x+2x + 2 is a factor of the polynomial.

Step 3: Polynomial Division

We divide x3−x+6x^3 - x + 6 by x+2x + 2 using synthetic or long division.

Using synthetic division with the root −2-2:

−210−16−24−61−230\begin{array}{r|rrr} -2 & 1 & 0 & -1 & 6 \\ & & -2 & 4 & -6 \\ \hline & 1 & -2 & 3 & 0 \\ \end{array}

So, the quotient is x2−2x+3x^2 - 2x + 3.

Step 4: Combine the factorization

Thus, we have:
x3−x+6=(x+2)(x2−2x+3)x^3 - x + 6 = (x + 2)(x^2 - 2x + 3)

Step 5: Check if quadratic can be factored further

To see if x2−2x+3x^2 - 2x + 3 can be factored further, we'll check if it has real roots by finding the discriminant:
Δ=b2−4ac=(−2)2−4(1)(3)=4−12=−8\Delta = b^2 - 4ac = (-2)^2 - 4(1)(3) = 4 - 12 = -8

Since the discriminant is negative ( −8-8 ), the quadratic x2−2x+3x^2 - 2x + 3 has no real roots and cannot be factored further over the reals.

Final Factorization

The final factorization of x3−x+6x^3 - x + 6 is:
x3−x+6=(x+2)(x2−2x+3)x^3 - x + 6 = (x + 2)(x^2 - 2x + 3)

This is the most simplified factorization over the real numbers.