The function below has at least one rational zero.
Use this fact to find all zeros of the function.
8 (x) = 4x^3 - 16x^2 + 9x+9
If there is more than one zero, separate them with commas. Write exact values, not decimal approximations.
To find the rational zeros of the function 8(x) = 4x^3 - 16x^2 + 9x + 9, we can use the Rational Root Theorem.
The Rational Root Theorem states that if a polynomial has a rational root p/q (where p and q are integers and q is not zero), then p must divide evenly into the constant term and q must divide evenly into the leading coefficient.
In this case, the constant term is 9 and the leading coefficient is 4.
The possible rational roots are therefore ±1, ±3, ±9, ±1/2, ±3/2, ±9/2.
By trying out the factors of 9 in the polynomial using synthetic division, we find that x = -3 is a root. Performing synthetic division by dividing (x + 3) into the polynomial gives:
4x^3 - 16x^2 + 9x + 9 = (x + 3)(4x^2 - 28x + 3)
Now we can solve the quadratic equation 4x^2 - 28x + 3 = 0 to find the remaining roots. Using the quadratic formula, we get:
x = (28 ± sqrt(784 - 48)) / 8
x = (28 ± sqrt(736)) / 8
x = (28 ± 8sqrt(46)) / 8
x = 7 ± sqrt(46)
Therefore, the zeros of the function 8(x) = 4x^3 - 16x^2 + 9x + 9 are x = -3, x = 7 + sqrt(46), and x = 7 - sqrt(46).