# SOLUTION: Polynomial function: a. List all possible zeros b. find all rational zeros c. list all factors! f(x) = x^3 + 5x^2 + 2x -8

Algebra ->  Algebra  -> Polynomials-and-rational-expressions -> SOLUTION: Polynomial function: a. List all possible zeros b. find all rational zeros c. list all factors! f(x) = x^3 + 5x^2 + 2x -8       Log On

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 Click here to see ALL problems on Polynomials-and-rational-expressions Question 77338: Polynomial function: a. List all possible zeros b. find all rational zeros c. list all factors! f(x) = x^3 + 5x^2 + 2x -8 Answer by Edwin McCravy(8909)   (Show Source): You can put this solution on YOUR website!``` Polynomial function: a. List all possible zeros b. find all rational zeros c. list all factors! f(x) = x³ + 5x² + 2x - 8 The rule is: If a polynomial arranged in descending order has any rational zeros, then they will be among the set of positive and negative fractions, each of whose numerator is a factor of the constant term, (i.e. the term with no vraiable), and whose denominator is a factor of the coefficient of the first term. The polynomial is so arranged. The constant term is -8, and the coefficient of the first term is 1, so if there are any rational zeros, they will be among these: ±1, ±2, ±4, ±8 Try the easiest one first, 1. If 1 is a solution, then (x - 1) will be a factor of f(x) using synthetic division we divide by (x - 1): 1|1 5 2 -8 | 1 6 8 1 6 8 0 Since the last number on the bottom row of the aynthetic division (the remainder) is 0, we have factored f(x) as f(x) = (x - 1)(x² + 6x + 8) We can now factor the trinomial in the second parentheses: f(x) = (x - 1)(x + 2)(x + 4) That's the list of factors, (x - 1), (x + 2), and (x + 4) Setting each of the factors = 0 we have x = 1, x = -2, and x = -4 as the three zeros. Here's the graph of f(x). Notice that it crosses the x-axis 3 times, once at each those three zeros. Edwin```