List all possible rational zeros of each function.
f(x)=4x^5+4x^2-9
Find all rational roots of each function
f(x)=2x^3+x^2+8x+4
second:
x^2(2x+1)+4(2x+1)
(2x+1)(x^2+4)
(2x+1)(x+j)(x-j)
so if the expression were equal zero, then
zeroes are -1/2, -j,+j where j= sqrt(-1)
so the only rational root is -1/2
To find the possible rational zeros of a function, we can use a technique called the Rational Root Theorem. According to the theorem, any rational root of a function can be expressed as the quotient of a factor of the constant term divided by a factor of the leading coefficient.
For the function f(x) = 4x^5 + 4x^2 - 9, the constant term is -9, and the leading coefficient is 4. We need to consider the factors of -9 and 4 to find the possible rational zeros.
The factors of -9 are ±1, ±3, and ±9.
The factors of 4 are ±1 and ±2.
Considering all possible combinations of these factors, the potential rational zeros for f(x) are:
±1/1, ±3/1, ±9/1, ±1/2, ±3/2, and ±9/2.
Now let's move on to the next question.
For the function f(x) = 2x^3 + x^2 + 8x + 4, the constant term is 4, and the leading coefficient is 2.
The factors of 4 are ±1 and ±2.
The factors of 2 are ±1.
Considering all possible combinations of these factors, the potential rational zeros for f(x) are:
±1/1, ±2/1, ±1/2, and ±2/2.
These are the possible rational zeros for each function. Note that these are potential zeros, and some or all of them may not actually be roots of the functions.