List all possible rational roots for the function
f(x) = 7x^4 - 5x^3 - 4x^2 + 9x + 119
Give your list in increasing order. Beside each possible rational root, type "yes" if it is a root and "no" if it is not a root.
I honestly have no real idea how to solve these problems.... So explanations would be great!
f(x) = 7x^4 - 5x^3 - 4x^2 + 9x + 119
Give your list in increasing order. Beside each possible rational root, type "yes" if it is a root and "no" if it is not a root.
I honestly have no real idea how to solve these problems.... So explanations would be great!
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Use the rational roots theorem. If p/q, p an q ≠ 0 integers with gcd = 1, is a root of f, then p divides 119 = 7 x 17 (the idependent term) and q divides 7 (the coefficient of the leading term). The positive divisors of 119 are 1, 7, 17 and 119; and those of 7 are 1 and 7. To find the possible positive rational roots we compute
1/1, 7/1, 17/1, 119/1
1/7, 7/7 = 1, 17/7, 119/7 = 17
So, the possible positive rational roots of f are 1, 7, 17, 119, 1/7, 17/7. And the possible negative rational roots are the symmetric of these numbers. In increasing order, the possible rational roots of f are
-119, -17, -7, -17/7, -1, 1/7, 1/7, 1, 17/7, 7, 17, 119
We can ensure no other rational is root of f. To know which of the above numbers (if any) are roots of f, you have to plug them in the expression of f, do the computations and see if the result is 0. I won't do this. Use a spreadsheet
1/1, 7/1, 17/1, 119/1
1/7, 7/7 = 1, 17/7, 119/7 = 17
So, the possible positive rational roots of f are 1, 7, 17, 119, 1/7, 17/7. And the possible negative rational roots are the symmetric of these numbers. In increasing order, the possible rational roots of f are
-119, -17, -7, -17/7, -1, 1/7, 1/7, 1, 17/7, 7, 17, 119
We can ensure no other rational is root of f. To know which of the above numbers (if any) are roots of f, you have to plug them in the expression of f, do the computations and see if the result is 0. I won't do this. Use a spreadsheet