SOLUTION: Is it possible for a polynomial function to have no rational zeroes but to have real zeros? If so, give an example. So, I was also confused on the difference between real and ratio

Question 1014628: Is it possible for a polynomial function to have no rational zeroes but to have real zeros? If so, give an example. So, I was also confused on the difference between real and rational zeros, does the rational root test have a play?
Found 2 solutions by stanbon, macston:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
Is it possible for a polynomial function to have no rational zeroes but to have real zeros? If so, give an example. So, I was also confused on the difference between real and rational zeros, does the rational root test have a play?
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sqrt(2) and sqrt(3) are Real but not Rational numbers.
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y = (x-sqrt(2))(x-sqrt(3))
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y = x^2 -(sqrt(2)+sqrt(3))x + sqrt(6)
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Cheers,
Stan H.
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Answer by macston(5194) (Show Source): You can put this solution on YOUR website!
.
Yes
.
x^2-3=0
.
roots (zeroes) at +/-
The roots are real, but not rational.
(The square root of three is not rational)
Real (not complex) roots are always x-intercepts
on the graph of the function. Imaginary roots are not.
.
On the graph below:
.
Red line below, has irrational real roots
.
Green line below, has imaginary (not real) roots,
and does not intercept x axis.
.

.
The rational roots test gives possible rational roots.
If there are rational roots,they are given by this test.
The test does not determine if any of the possibilities
are actually roots.
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