Thank you for visiting Use the Rational Root Theorem to factor the following polynomial equation x 5 8x 4 39x 3 326x 2 140x 1176 0. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
By applying the Rational Root Theorem, you can factorize the given polynomial as (x + 3)(x - 2)(x - 4)(x^2 - 4x + 2). This theorem helps identify potential rational roots of the polynomial.
In this problem, we are using the Rational Root Theorem to factor the given polynomial equation, x^5 - 8x^4 - 39x^3 + 326x^2 + 140x - 1176.
The Rational Root Theorem states that if a polynomial has a rational root, p/q, then p is a factor of the constant term, and q is a factor of the leading coefficient.
In our case, the constant term is -1176 and the leading coefficient is 1. So, potential rational roots for this polynomial are p/q such that p is a factor of -1176 and q is a factor of 1.
By trial-and-error, you can find that x = -3, x = 2, and x = 4 are roots of the polynomial.
Therefore, the polynomial factors as (x + 3)(x - 2)(x - 4)(x^2 - 4x + 2). The last factor is a quadratic and does not have rational roots, so the factorization is complete.
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