Please check the first and help with the second-thank you
Find all possible rational roots of f(x) = 2x^4 - 5x^3 + 8x^2 + 4x+7
1.I took the constant which is 7 and the leading coefficient which is 2 and factored them
7 factored would be 7,2
2 factored would be 2,1
used 7 as numerator 2 as denominator came up with 1/2,1/7/2, 7 Would that be correct?
2.I don't get this one find number of possible positive and negative real roots of f(x) = x^4-x^3+2x^2 + x-5
I think there are 3 sign changes so there are 3 positive but I get really confused on doing the negative real zeros would I rewrite for negative f(x) = -x^4 -(-x^3) -2x^2 -(x-5) so there would be 2negative real zeros
I don't get this one
No, the numbers you state would be potential rational roots.
You still have to check to see if they actually work
e.g.
f(1/2) = 2(1/2)^4 - 5(1/2)^3 + 8(1/2)^2 + 4(1/2) + 7 = 10.5 ≠ 0
f(1) = 2 - 5 + 8 + 4 + 7 ≠ 0
etc.
none of them work. A quick look at "Wolfram" shows that there are no real roots at all.
http://www.wolframalpha.com/input/?i=2x%5E4+-+5x%5E3+%2B+8x%5E2+%2B+4x%2B7+%3D0
For negative zeros, substitute -x for x, but do it carefully, using the exact same coefficients as originally:
(-x)^4 - (-x)^3 + 2(-x)^2 + (-x) - 5
or
x^4 + x^3 + 2x^2 - x - 5
There's only one change of sign there, so at most one negative root.
To find the possible rational roots of a polynomial equation, such as f(x) = 2x^4 - 5x^3 + 8x^2 + 4x + 7, you can use the Rational Root Theorem. The theorem states that any rational root of the polynomial equation will be of the form p/q, where p is a factor of the constant term (in this case, 7) and q is a factor of the leading coefficient (in this case, 2).
1. For the first question, you correctly identified the constant term as 7 and the leading coefficient as 2. You factored 7 and 2 to get possible factors for p and q. However, you made a mistake in using the numerator as the factor of the constant term and the denominator as the factor of the leading coefficient. Instead, you should consider all the possible combinations of p/q using the factors of 7 as the numerator and the factors of 2 as the denominator. Therefore, the possible rational roots are ±1, ±7/2.
2. To determine the number of possible positive and negative real roots of the polynomial f(x) = x^4 - x^3 + 2x^2 + x - 5, you can use the Descartes' Rule of Signs.
First, count the sign changes in the original polynomial: "+ - + + -". There are two sign changes in this case.
Next, we count the sign changes for the polynomial with its terms arranged in descending order of degree, considering negative values of x. So let's rewrite the polynomial as f(-x) = (-x)^4 - (-x)^3 + 2(-x)^2 + (-x) - 5, which simplifies to f(-x) = x^4 + x^3 + 2x^2 - x - 5.
Counting the sign changes in f(-x) gives us: "+ + + - -". There are three sign changes for this polynomial.
According to Descartes' Rule of Signs, the number of positive real roots is equal to the number of sign changes in the original polynomial (2 in this case) or fewer by an even number. So there can be either 0 or 2 positive real roots.
Similarly, the number of negative real roots is equal to the number of sign changes in f(-x) (3 in this case) or fewer by an even number. So there can be either 0, 2, or 4 negative real roots.
In summary, there can be 0, 2, or 4 positive real roots, and there can be 0, 2, or 4 negative real roots.