i dont understand what rational zeros are.
Find the rational zeros of f(x)=x^4-4x^3+x^2+16x-20.
This question would come in a unit on polynomial functions after the introduction of the Rational Root Theorem. The student would be expected to solve it this way:
A rational zero is a rational number, say p/q to lowest terms, that makes f(p/q) = 0.
The rational zero theorem says that p is a factor of the constant term of f, and q is a factor of the leading coefficient of f.
So all the possible rational zeros of f are:
factors of -20 ±1, ±2, ±4, ±5, ±10, ±20
--------------- => --------------------------------
factors of 1 ±1
=> ±1, ±2, ±4, ±5, ±10, and ±20 are the only possible rational zeros.
Now they need to be tested:
2 | 1 -4 1 16 -20
--- 2 -4 -6 20
------------------
1 -2 -3 10 | 0
----
So f(x) = (x - 2) (x^3 - 2 x^2 - 3 x + 10).
2 | 1 -2 -3 10
--- 2 0 -6
-------------
1 0 -3 | X => no double zero
----
-2 | 1 -2 -3 10
---- -2 8 -10
------------
1 -4 5 | 0
----
So f(x) = (x + 2) (x - 2) (x^2 - 4 x + 5).
For x^2 - 4 x + 5 = 0,
b^2 - 4ac = 16 - 20 = -4,
This question would come in a unit on polynomial functions after the introduction of the Rational Root Theorem. The student would be expected to solve it this way:
A rational zero is a rational number, say p/q to lowest terms, that makes f(p/q) = 0.
The rational zero theorem says that p is a factor of the constant term of f, and q is a factor of the leading coefficient of f.
So all the possible rational zeros of f are:
factors of -20 ±1, ±2, ±4, ±5, ±10, ±20
--------------- => --------------------------------
factors of 1 ±1
=> ±1, ±2, ±4, ±5, ±10, and ±20 are the only possible rational zeros.
Now they need to be tested:
2 | 1 -4 1 16 -20
--- 2 -4 -6 20
------------------
1 -2 -3 10 | 0
----
So f(x) = (x - 2) (x^3 - 2 x^2 - 3 x + 10).
2 | 1 -2 -3 10
--- 2 0 -6
-------------
1 0 -3 | X => no double zero
----
-2 | 1 -2 -3 10
---- -2 8 -10
------------
1 -4 5 | 0
----
So f(x) = (x + 2) (x - 2) (x^2 - 4 x + 5).
For x^2 - 4 x + 5 = 0,
b^2 - 4ac = 16 - 20 = -4,
x = (-(-4) ± sqrt(-4))/2 = 2 ± i, complex conjugates.
So f(x) has only two rational zeros: x = ±2
Notes: Rational implies Real. Real does not imply Rational. The fourth degree polynomial function
So f(x) has only two rational zeros: x = ±2
Notes: Rational implies Real. Real does not imply Rational. The fourth degree polynomial function
g(x) = 5 (x ±√2) (x ±√3) = (x^2 - 2) (x^2 - 3) has 4 real zeros but none of them are rational.