Using the rational root theorem, list out all possible/candidate rational roots of f, of, x, equals, 11, x, cubed, minus, 14, plus, x, minus, 2, x, to the power 4 , plus, 17, x, squaredf(x)=11x 3 −14+x−2x 4 +17x 2 . 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, 11, x, cubed, minus, 14, plus, x, minus, 2, x, to the power 4 , plus, 17, x, squaredf(x)=11x 3 −14+x−2x 4 +17x 2 . 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, of a polynomial must have p as a factor of the constant term and q as a factor of the leading coefficient.
The polynomial given is f(x) = -2x^4 + 11x^3 + 17x^2 + x - 14.
The constant term is -14 and the leading coefficient is -2.
The factors of -14 are ±1, ±2, ±7, ±14 and the factors of -2 are ±1, ±2.
So, the possible rational roots are all the combinations of these factors, which are:
±1/1, ±2/1, ±7/1, ±14/1, ±1/2, ±2/2, ±7/2, ±14/2
Simplifying these gives:
±1, ±2, ±7, ±14, ±1/2, ±7/2
So, the possible rational roots of the polynomial are ±1, ±2, ±7, ±14, ±1/2, ±7/2.
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