Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to discard any rational zeros that are obviously not zeros of the function. (Enter your answers as a comma-separated list.)
f(x) = 9x3 − 30x2 + 37x − 26
x =
If X= 3
How the is the perimeter of the rectangle 2(6X) + 2(W)=?
9x^3 - 30x^2 + 37x - 26 = 0
A little poking around with synthetic division show that
f(x) = (x-2)(9x^2-12x+13)
So, the only real root is x=2
To find the zeros of the function f(x) = 9x^3 - 30x^2 + 37x - 26, we can use the Rational Zero Theorem to generate a list of possible rational zeros.
The Rational Zero Theorem states that if a rational number p/q is a zero of the polynomial function f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, then p must be a divisor of a_0 and q must be a divisor of a_n.
In this case, the leading coefficient is 9 and the constant term is -26. So, the possible rational zeros are the divisors of -26 divided by the divisors of 9.
The divisors of -26 are: ±1, ±2, ±13, ±26.
The divisors of 9 are: ±1, ±3, ±9.
By testing these possible zeros in the function using a graphing utility, we can check which ones are actual zeros of the function.
Using a graphing utility, we find that the zeros of the function f(x) = 9x^3 - 30x^2 + 37x - 26 are:
x = -2, 2/3, and 13/3.
Hence, the zeros of the function are x = -2, x = 2/3, and x = 13/3.
To find the zeros of the function f(x) = 9x^3 - 30x^2 + 37x - 26, we can use the Rational Root Theorem to determine a list of possible rational zeros.
The Rational Root Theorem states that if a rational number p/q, where p is a factor of the constant term (in this case -26) and q is a factor of the leading coefficient (in this case 9), is a zero of the polynomial, then p will be a factor of the constant term and q will be a factor of the leading coefficient.
Factors of 9: ±1, ±3, ±9
Factors of 26: ±1, ±2, ±13, ±26
Using the Rational Root Theorem, the possible rational zeros are: ±1/1, ±2/1, ±3/1, ±13/1, ±26/1 .
To check which of these possible zeros are actually zeros of the function, we can use a graphing utility to graph the function and observe where the graph intersects the x-axis. This will give us a visual indication of the zeros.
Graphing the function f(x) = 9x^3 - 30x^2 + 37x - 26, we can see that the function intersects the x-axis at two distinct points.
The zeros of the function are approximately x = 1.9143, x = 1.0476, and x = -0.9619.
Therefore, the zeros of the function f(x) = 9x^3 - 30x^2 + 37x - 26 are x = 1.9143, x = 1.0476, and x = -0.9619.