find the complex zeros of the polynomial function. write F in the factored form. f(X)=x^3-7x^2+20x-24
use the complex zeros to write f in factored form.
f(x)=
(reduce fractions and simplify roots)
To find the complex zeros of the polynomial function and write it in factored form, we can use the Rational Root Theorem and synthetic division.
First, let's list all the possible rational roots (p/q) of the polynomial by using the Rational Root Theorem. The Rational Root Theorem states that if a polynomial function has a rational root (p/q), then p is a factor of the constant term (in this case, -24), and q is a factor of the leading coefficient (in this case, 1).
The factors of -24 are:
±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24
The factors of 1 are:
±1
So, the possible rational roots are:
±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24
To find the complex zeros, we will use synthetic division for each possible rational root:
For the polynomial f(x) = x^3-7x^2+20x-24:
1. Testing for x = 1:
Perform synthetic division with the coefficients of f(x) and test if the remainder is 0.
1 │ 1 -7 20 -24
│ 1 -6 14
└───────────
1 -6 14 -10
The remainder is -10, meaning x = 1 is not a root of f(x).
2. Testing for x = -1:
Perform synthetic division with the coefficients of f(x) and test if the remainder is 0.
-1 │ 1 -7 20 -24
│ -1 8 -28
└────────────
1 -8 28 -52
The remainder is -52, meaning x = -1 is not a root of f(x).
3. Testing for x = 2:
Perform synthetic division with the coefficients of f(x) and test if the remainder is 0.
2 │ 1 -7 20 -24
│ 2 -10 20
└────────────
1 -5 10 -4
The remainder is -4, meaning x = 2 is not a root of f(x).
4. Testing for x = -2:
Perform synthetic division with the coefficients of f(x) and test if the remainder is 0.
-2 │ 1 -7 20 -24
│ -2 18 -76
└─────────────
1 -9 38 -100
The remainder is -100, meaning x = -2 is not a root of f(x).
5. Testing for x = 3:
Perform synthetic division with the coefficients of f(x) and test if the remainder is 0.
3 │ 1 -7 20 -24
│ 3 -12 24
└────────────
1 -4 8 0
The remainder is 0, meaning x = 3 is a root of f(x).
Now we have found one root, x = 3. To find the other complex roots, we need to divide f(x) by (x - 3) using either long division or synthetic division.
Performing synthetic division with the coefficients of f(x) and the root x = 3:
3 │ 1 -4 8
│ 3 -3
└─────────
1 -1 5
The result is the quadratic equation x^2 - x + 5. To find its complex roots, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
For x^2 - x + 5, a = 1, b = -1, and c = 5.
x = (-(-1) ± √((-1)^2 - 4(1)(5))) / (2(1))
x = (1 ± √(1 - 20)) / 2
x = (1 ± √(-19)) / 2
x = (1 ± i√19) / 2
Therefore, the complex zeros of the polynomial function f(x) = x^3 - 7x^2 + 20x - 24 are:
x = 3, x = (1 ± i√19) / 2
To write f(x) in factored form, we can use the roots we found:
f(x) = (x - 3)(x - (1 + i√19)/2)(x - (1 - i√19)/2)
Make sure to further simplify and reduce fractions if needed.
try factors of 24
f(1) = 1 - 7 + 20 - 24 ≠0
f(-1) = -1 -7 - 20 - 24 ≠ 0
f(2) = ≠0
f(-2) ≠ 0
f(3) = 27 - 63 + 60 - 24 = 0
So (x-3) is a factor
Using synthetic division, I got
x^3-7x^2+20x-24 = (x-3)(x^2 -4x + 8)
Solving the 2nd part:
x^2 - 4x + .... = -8 + ....
x^2 - 4x + 4 = -8+4
(x-2)^2 = -4
x-2 = ± 2i
x = 2 ± 2i
f(x) = (x-3)(x^2 -4x + 8)