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We can factor the polynomial $x^5-3x^4-23x^3+51x^2+94x-120$ using the rational root theorem, which guarantees that for a polynomial of the form $a_nx^n+a_{n-1}x^{n-1}+\dots+a_0$ there is a rational root of the form $\pm\frac{p}{q}$, where $p$ belongs to the divisors of the constant term $a_0$, and $q$ belongs to the divisors of the leading coefficient $a_n$. List all divisors $p$ of the constant term $a_0$, which equals $-120$
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$1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120$
Learn how to solve polynomial factorization problems step by step online. Factor by completing the square (-x^4-22x^3134x^2280x+-996)/(x^5-3x^4-23x^351x^294x+-120). We can factor the polynomial x^5-3x^4-23x^3+51x^2+94x-120 using the rational root theorem, which guarantees that for a polynomial of the form a_nx^n+a_{n-1}x^{n-1}+\dots+a_0 there is a rational root of the form \pm\frac{p}{q}, where p belongs to the divisors of the constant term a_0, and q belongs to the divisors of the leading coefficient a_n. List all divisors p of the constant term a_0, which equals -120. Next, list all divisors of the leading coefficient a_n, which equals 1. The possible roots \pm\frac{p}{q} of the polynomial x^5-3x^4-23x^3+51x^2+94x-120 will then be. Trying all possible roots, we found that 5 is a root of the polynomial. When we evaluate it in the polynomial, it gives us 0 as a result.