Using the remainder and factor theorems, factorize the polynomial.

x³ + 10x² - 37x + 26

Let, f(x) = x³ + 10x² - 37x + 26.

Substituting, x = 1 in f(x) we get :

f(1) = (1)³ + 10(1)² - 37(1) + 26

= 1 + 10 - 37(1) + 26

= 37 - 37

= 0.

Since, f(1) = 0, thus (x − 1) is a factor of f(x).

Dividing, x³ + 10x² - 37x + 26 by (x - 1), we get :

$\begin{array}{l} \phantom{x - ]3)}{x^2 + 11x - 26} \ x - 1\overline{\smash{\big)}x^3 + 10x^2 - 37x + 26} \ \phantom{x - 2}\underline{\underset{-}{}x^3 \underset{+}{-}x^2} \ \phantom{{x - 2}x^,,,3-}11x^2 - 37x \ \phantom{{x -l2}fx^3]\space}\underline{\underset{-}{+}11x^2 \underset{+}{-}11x} \ \phantom{{x - 2]euo[ki]}x^3okk\space}{-26x + 26} \ \phantom{{x - 2}x^3o;llk]lmttk\space}\underline{\underset{+}{-}26x\underset{-}{+}26} \ \phantom{{x - 2}{x^,jo-k2x^mm2\space}{-9x}}\times \end{array}$

∴ x³ + 10x² - 37x + 26 = (x - 1)(x² + 11x - 26)

= (x - 1)(x² + 13x - 2x - 26)

= (x - 1)[x(x + 13) - 2(x + 13)]

= (x - 1)(x + 13)(x - 2).

Hence, x³ + 10x² - 37x + 26 = (x - 1)(x + 13)(x - 2).