What is the number of distinct possible rational roots of the polynomial P(x)=5x2+19x−4
i know that the actual roots of the polynomial are ±1,±1/5,±2,±2/5,±4,±4/5 through finding the rational roots but I am confused on what the question is asking by saying the possible distance.
The choices are 6,12,5, or 10
it's a 2nd order polynomial ... there are only two roots
your list of "actual roots" is the answer to the question
... how many are there?
HUH ?
if
y = 5 x^2 + 19 x - 4
the roots are when y = 0
5 x^2 + 19 x - 4 = 0
(5x-1)(x+4) = 0
x = 1/5 and x = -4
the polynomial that has all the roots you gave is
(x-1)(+1)(5x-1)(5x+1)(x-2)(x+2) (5x-2)(5x+2) .....
Typo or something ?
±1,±1/5,±2,±2/5,±4,±4/5 are all the roots I came to the conclusion of but the choices they are giving me are 6,12,5, or 10 to find f distinct possible rational roots of the polynomial
There are only two roots of a polynomial that has x^2 as the highest power of x in it.
That does not make sense to me at all
Well you have a typo or something.
A polynomial like
a x^2 + b x + c
can only cross the x axis twice, maximum. (If it never crosses, roots have imaginary parts)
like if polynomial is x^2 + 10
that is zero when x = +/- sqrt( -10) which is +/- i sqrt (10)
if x^2 + x + 10
then
[-1 +/-sqrt(-39) ] / 2
the question is asking about POSSIBLE rational roots
for the general equation ... f(x) = a x^2 + b x + c
... the POSSIBLE rational roots are ... (factors of c) / (factors of a)
LOL, thanks R Scott !
The question is asking for the number of distinct possible rational roots of the given polynomial, P(x) = 5x^2 + 19x - 4.
To determine the possible rational roots, we can use the Rational Root Theorem. According to this theorem, if a rational number p/q is a root of the polynomial, it must satisfy two conditions: p is a divisor of the constant term (-4), and q is a divisor of the leading coefficient (5).
Let's list the factors of -4: ±1, ±2, ±4. These are the possible values for p.
Now let's list the factors of 5: ±1, ±5. These are the possible values for q.
To find the possible rational roots, we construct all possible fractions by dividing each possible value of p by each possible value of q.
So the possible rational roots are: ±1, ±1/5, ±2, ±2/5, ±4, ±4/5.
There are 12 possible rational roots.
Therefore, the correct answer is 12.