use rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers.
f(x)=x^3-8x^2-31x-22
**please show work**
all rational roots will be factors of 22
trying some, see that they are -1, -2, 11
To find the real zeros of a polynomial function using the Rational Zeros Theorem, you need to consider all the possible rational zeros that can be formed by taking the factors of the constant term (the coefficient of the independent term) divided by the factors of the leading coefficient (the coefficient of the highest power term).
In the given polynomial function f(x) = x^3 - 8x^2 - 31x - 22, the leading coefficient is 1, and the constant term is -22. Therefore, the factors of the constant term are ±1, ±2, ±11, and ±22, while the factors of the leading coefficient are ±1.
Now, we can create a list of all possible rational zeros by taking each factor of the constant term and dividing it by each factor of the leading coefficient.
Possible rational zeros:
±1/1, ±2/1, ±11/1, ±22/1
±1, ±2, ±11, ±22
To determine which of these possible rational zeros are actual zeros of the function, you can utilize synthetic division or long division to check each value individually.
Let's check each possible rational zero one by one using synthetic division.
For 1:
1 | 1 -8 -31 -22
| 1 -7 -38
|------------------
1 -7 -38 -60
For -1:
-1 | 1 -8 -31 -22
| -1 9 22
|-----------------
1 -9 -22 0
For 2:
2 | 1 -8 -31 -22
| 2 -12 -38
|-----------------
1 -6 -43 -60
For -2:
-2 | 1 -8 -31 -22
| -2 20 22
|-----------------
1 -10 -11 0
For 11:
11 | 1 -8 -31 -22
| 11 30 -2
|-----------------
1 3 -1 -24
For -11:
-11 | 1 -8 -31 -22
| -11 114 -83
|------------------
1 3 83 -105
For 22:
22 | 1 -8 -31 -22
| 22 300 -22
|------------------
1 14 -31 -44
For -22:
-22 | 1 -8 -31 -22
| -22 540 -180
|-------------------
1 -30 509 -202
From the synthetic division, we observe that f(1) = 0, f(-1) = 0, and f(-2) = 0, which implies that 1, -1, and -2 are roots of the polynomial.
By factoring f(x) using these zeros, we can write:
f(x) = (x - 1)(x + 1)(x + 2)
Thus, the real zeros of the polynomial function f(x) = x^3 - 8x^2 - 31x - 22 are x = 1, x = -1, and x = -2. It can be factored as f(x) = (x - 1)(x + 1)(x + 2).