The function below has at least one rational zero.
Use this fact to find all zeros of the function.
g(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 zero(s) of the function g(x) = 4x^3 - 16x^2 + 9x + 9, we can use the Rational Root Theorem. According to the theorem, any rational zero of the function must be a factor of the constant term 9 divided by a factor of the leading coefficient 4.
The factors of 9 are ±1, ±3, ±9 and the factors of 4 are ±1, ±2, ±4. So the possible rational zeros are ±1, ±3, ±9, ±1/2, ±3/2, ±9/2.
Let's try these possible zeros by synthetic division:
g(1) = 4(1)^3 - 16(1)^2 + 9(1) + 9 = 4 - 16 + 9 + 9 = 6 (not zero)
g(-1) = 4(-1)^3 - 16(-1)^2 + 9(-1) + 9 = -4 - 16 - 9 + 9 = -20 (not zero)
g(3) = 4(3)^3 - 16(3)^2 + 9(3) + 9 = 108 - 144 + 27 + 9 = 0
Since g(3) = 0, x = 3 is a zero of the function. By performing synthetic division with x = 3, we obtain:
g(x) = (x - 3)(4x^2 - 4x - 3)
To find the remaining zeros, we solve the quadratic equation 4x^2 - 4x - 3 = 0 using the quadratic formula:
x = [-(-4) ± sqrt((-4)^2 - 4(4)(-3))] / (2*4)
x = [4 ± sqrt(16 + 48)] / 8
x = [4 ± sqrt(64)] / 8
x = [4 ± 8] / 8
This gives us two more zeros:
x = (4 + 8) / 8 = 12 / 8 = 3/2
x = (4 - 8) / 8 = -4 / 8 = -1/2
Therefore, the zeros of the function g(x) = 4x^3 - 16x^2 + 9x + 9 are x = 3, x = 3/2, and x = -1/2.