Using the rational zeros theorem to find all zeros of a polynomial

The function below has at least one rational zero. Use this fact to find all zeros of the function

g(x)=5x©ù-28©ø-48x©÷-8x+7

if more than one zero, separate with commas. Write exact values, not decimal approximations

+-(1,7)/(1,5)=+-1/(5,1)

Assuming you meant

g(x) = (5x^3 - 28x^2 - 48x)/(-8x+7)

it's not a polynomial. The possible rational zeroes of the numerator include all the fractions whose numerator is a factor of 48 and whose denominator is 1 or 5.

The only rational root of that is 0.

Would you care to repost the question, using ^ for exponents, as I did above?

To find the rational zeros of a polynomial, we can use the Rational Zeros Theorem. The Rational Zeros Theorem states that if a polynomial function has a rational zero of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient, then p is a factor of the constant term and q is a factor of the leading coefficient.

Looking at the polynomial function g(x) = 5x³ - 28x² - 48x² - 8x + 7, we can identify the leading coefficient as 5 and the constant term as 7.

Factors of the constant term (7) are ±1 and ±7. Factors of the leading coefficient (5) are ±1 and ±5.

Now, we can find possible rational zeros by forming all possible fractions using the factors we just found. This gives us the possible rational zeros as ±1, ±7, ±1/5, and ±7/5.

Therefore, the possible rational zeros for the function g(x) are ±1, ±7, ±1/5, and ±7/5.

In order to determine which of these possible rational zeros are actually zeros of the function, we can use a process called synthetic division or evaluate the function at each of the possible rational zeros.

Let's evaluate the function g(x) at each of the possible rational zeros:
- g(1) = 5(1)³ - 28(1)² - 48(1)² - 8(1) + 7 = -74 (not zero)
- g(-1) = 5(-1)³ - 28(-1)² - 48(-1)² - 8(-1) + 7 = -28 (not zero)
- g(7) = 5(7)³ - 28(7)² - 48(7)² - 8(7) + 7 = 0 (zero!)
- g(-7) = 5(-7)³ - 28(-7)² - 48(-7)² - 8(-7) + 7 = -666 (not zero)
- g(1/5) = 5(1/5)³ - 28(1/5)² - 48(1/5)² - 8(1/5) + 7 = -276/25 (not zero)
- g(-1/5) = 5(-1/5)³ - 28(-1/5)² - 48(-1/5)² - 8(-1/5) + 7 = -184/25 (not zero)
- g(7/5) = 5(7/5)³ - 28(7/5)² - 48(7/5)² - 8(7/5) + 7 = 0 (zero!)
- g(-7/5) = 5(-7/5)³ - 28(-7/5)² - 48(-7/5)² - 8(-7/5) + 7 = -2744/125 (not zero)

From the evaluations, we can see that the zeros of the function g(x) are x = 7 and x = 7/5.

The zeros of the function g(x) are 7 and 7/5.