use the rational zero theorem to list the potential rational zeros of the polynomial function. Do not attempt to find the zeros.
f(x)=2x^7-x^5+8x^4+52
if p/q is a rational root, then p divides 52 and 1 divides 2
p: ±1 ±2 ±4 ±13 ±26 ±52
q: ±1 ±2
now, list possible p/q values
To list the potential rational zeros of a polynomial function using the Rational Zero Theorem, you will need to look at the factors of the constant term divided by the factors of the leading coefficient.
In the given polynomial function f(x) = 2x^7 - x^5 + 8x^4 + 52, the constant term is 52 and the leading coefficient is 2.
Step 1: Find the factors of the constant term (52)
The factors of 52 are:
±1, ±2, ±4, ±13, ±26, ±52
Step 2: Find the factors of the leading coefficient (2)
The factors of 2 are:
±1, ±2
Step 3: Write down all the possible combinations of the factors of the constant term divided by the factors of the leading coefficient:
±1/±1, ±1/±2, ±2/±1, ±2/±2, ±4/±1, ±4/±2, ±13/±1, ±13/±2, ±26/±1, ±26/±2, ±52/±1, ±52/±2
Now you have listed all the potential rational zeros of the polynomial function f(x) = 2x^7 - x^5 + 8x^4 + 52 based on the Rational Zero Theorem. These potential zeros are the values that you can try as potential solutions to the equation to check if they are actual roots.