Q. \(x^4 – 8x^3 + 13x^2 – 24x + 9\) factorization.

Problem

Factor the polynomial

\[ x^4 – 8x^3 + 13x^2 – 24x + 9. \]

Solution — step by step

1. Assume the quartic factors as a product of two quadratics with real coefficients:

   \[ (x^2 + a x + b)(x^2 + c x + d). \]

2. Expand the product to express coefficients in terms of \(a,b,c,d\):

   \[ (x^2 + a x + b)(x^2 + c x + d) = x^4 + (a + c)x^3 + (ac + b + d)x^2 + (ad + bc)x + bd. \]

3. Compare coefficients with the given polynomial \(x^4 – 8x^3 + 13x^2 – 24x + 9\). This yields the system:

   \[
   \begin{cases}
   a + c = -8, \\[4pt]
   ac + b + d = 13, \\[4pt]
   ad + bc = -24, \\[4pt]
   bd = 9.
   \end{cases}
   \]

4. Use \(bd = 9\). The integer factor pairs of 9 are \((1,9), (3,3), (9,1)\) and their negatives. Because the constant term is positive and the cubic coefficient is negative, try positive values for \(b\) and \(d\). Try \(b = 3\) and \(d = 3\).

5. With \(b = 3\) and \(d = 3\), the second equation gives:

   \[ ac + 3 + 3 = 13 \quad\text{so}\quad ac = 7. \]

   The first equation gives:

   \[ a + c = -8. \]

   Thus \(a\) and \(c\) are numbers whose sum is \(-8\) and product is \(7\).

6. Find \(a\) and \(c\) by solving the quadratic equation whose roots are \(a\) and \(c\):

   \[ t^2 – (a+c)t + ac = 0 \]

   so

   \[ t^2 + 8t + 7 = 0. \]

   Solve:

   \[ t = \frac{-8 \pm \sqrt{64 – 28}}{2} = \frac{-8 \pm \sqrt{36}}{2} = \frac{-8 \pm 6}{2}. \]

   Thus \(t = -1\) or \(t = -7\). Therefore \((a,c) = (-1,-7)\) in some order.

7. Check the coefficient of \(x\): with \(a = -1\), \(c = -7\), \(b = d = 3\),

   \[ ad + bc = (-1)\cdot 3 + 3\cdot(-7) = -3 -21 = -24, \]

   which matches the given coefficient. So the choice is consistent.

8. Write the factorization using these values:

   \[ x^4 – 8x^3 + 13x^2 – 24x + 9 = (x^2 – x + 3)(x^2 – 7x + 3). \]

9. Optional verification: multiply the factors to confirm they expand to the original polynomial:

   \[
   (x^2 – x + 3)(x^2 – 7x + 3)
   = x^4 – 8x^3 + 13x^2 – 24x + 9.
   \]

Final factorization:

\[ x^4 – 8x^3 + 13x^2 – 24x + 9 = (x^2 – x + 3)(x^2 – 7x + 3). \]