Using the rational root theorem, list out all possible/candidate rational roots of f, of, x, equals, 18, x, squared, minus, 3, x, cubed, plus, 7, plus, 17, xf(x)=18x 2 −3x 3 +7+17x. Express your answer as integers or as fractions in simplest form. Use commas to separate.
Question
Using the rational root theorem, list out all possible/candidate rational roots of f, of, x, equals, 18, x, squared, minus, 3, x, cubed, plus, 7, plus, 17, xf(x)=18x 2 −3x 3 +7+17x. Express your answer as integers or as fractions in simplest form. Use commas to separate.
Solution
The Rational Root Theorem states that any rational root, p/q (where p is a factor of the constant term and q is a factor of the leading coefficient), of a polynomial equation
f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_2x^2 + a_1x + a_0 = 0
can be expressed as a factor of the constant term divided by a factor of the leading coefficient.
Given the polynomial f(x) = -3x^3 + 18x^2 + 17x + 7, the constant term is 7 and the leading coefficient is -3.
The factors of 7 (constant term) are: ±1, ±7.
The factors of -3 (leading coefficient) are: ±1, ±3.
Therefore, the possible rational roots of the polynomial f(x) = -3x^3 + 18x^2 + 17x + 7, according to the Rational Root Theorem, are:
±1/1, ±7/1, ±1/3, ±7/3.
Simplifying these fractions, we get:
±1, ±7, ±1/3, ±7/3.
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