use rational zeros theorem to find all the real zeros in the polynomial function. use zeros to factor f over real numbers.
f(x)=x^3+10x^2-13x-22
find real zeros of f?
use real zeros to factor f?
I suggest you start by using the rational roots theorem to find at least one root. You will learn more by making the effort yourself.
If I understood it I would not have to ask for help! I got it wrong in the 1st place and wanted to see the right answer. Thank you!!
The theorem says that if there is a rational root, it is (in this case) a factor of 22, which are one or more ofthese:
+/1, +/-2, +/-11 or +/-22.
See if any of those work.
thank you!
x = 2 is a solution and (x-2) is a factor.
The other factor is
(x^3+10x^2-13x-22)/(x-2)
Polynomial long division will reduce that to a quadratic.
6x^3+25x^2-24x+5
To find the real zeros of a polynomial function and factor it over real numbers, we can use the Rational Zeros Theorem.
The Rational Zeros Theorem states that if a rational number P/Q is a zero of a polynomial function, then P is a factor of the constant term, and Q is a factor of the leading coefficient.
For the polynomial function f(x) = x^3 + 10x^2 - 13x - 22, let's first list all the possible rational zeros. The constant term is -22, and the leading coefficient is 1.
The factors of -22 are ±1, ±2, ±11, and ±22.
The factors of 1 (leading coefficient) are ±1.
Therefore, the possible rational zeros are:
±1, ±2, ±11, ±22.
Now, we will use synthetic division or long division to check which of these possible zeros are actually the real zeros of f(x).
Let's start with one of the possible zeros, for example, x = 1:
Using synthetic division:
1 | 1 10 -13 -22
| 1 11 -2
|___________________
1 11 -2 -24
Since we obtained a remainder of -24, which is not zero, x = 1 is not a zero of f(x).
We can repeat this process for the remaining possible zeros:
For x = -1:
-1 | 1 10 -13 -22
| -1 -9 22
|___________________
1 9 0
For x = 2:
2 | 1 10 -13 -22
| 2 24 22
|___________________
1 12 11
For x = -2:
-2 | 1 10 -13 -22
| -2 -16 58
|___________________
1 8 36
For x = 11:
11 | 1 10 -13 -22
| 11 210 287
|___________________
1 121 275
For x = -11:
-11 | 1 10 -13 -22
| -11 0 143
|___________________
1 -1 121
From these results, we can see that x = -1 is a real zero of f(x).
To factor f(x) using this real zero, we can now use the factor theorem. Since x = -1 is a zero, (x+1) is a factor of f(x).
By dividing f(x) by (x+1), we get:
(x+1)(x^2 + 9x - 22)
Therefore, the factored form of f(x) over real numbers is:
f(x) = (x+1)(x^2 + 9x - 22).
The real zeros of the polynomial function f(x) are x = -1. And the factored form of f(x) over real numbers is (x+1)(x^2 + 9x - 22).