Question 172978This question is from textbook ALGEBRA 2
: f(x) = x4 - 2x3 + 3x - 4 using the rational zero theorem the numbers after the x is to the power.
f(x) = 2x3 - x2 + 5x + 6
f(x) = x3 - 4x2 - 7x + 10
f(x) = x4 + 4x3 + x2 - 8x - 6
from a worksheet lesson 6.6
This question is from textbook ALGEBRA 2
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website! If a polynomial function, written in descending order, has integer coefficients, then any rational zero must be of the form ± p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
So, the lead coefficient is 1, meaning that the only factor of the lead coefficient is 1, and the factors of the constant term are 1, 2, and 4. Therefore the possible rational zeros are:
± 1
± 1/2
± 1/4
Now that you have several possibilities, you need to test until you find a rational zero, or have tested them all and find that there is no rational zero. For any polynomial p(x), if a is a zero, then p(x)/(x-a) will result in a quotient and a zero remainder. So replace a with each of the possible zeros and perform polynomial long division to determine if you have a zero remainder. If you find one, you can then test the quotient for rational zeros. If none of the possible rational zeros results in a zero remainder, then the polynomial has no rational zeros. Remember when performing polynomial long division (or synthetic division which is really the same thing) to put in a placeholder for any missing terms. In your first problem, you need a  place holder.
If you have a problem performing polynomial long division or synthetic division, you can also check by substituting the possible zero into the original polynomial and doing the resulting arithmetic. If it is a zero, the result will be zero, otherwise it isn't a zero.
In the case of your first problem, none of the possibilities is actually zero of the polynomial, so your first problem has no rational zeros. However, you should NOT take my word for this. You need to test each of the possibilities and convince yourself that none of them work.
All four of these problems are done in the same way.
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