Factoring a Polynomial
The polynomial given in the image is:
f(x) = x^3 - 9x^2 + 26x - 24
To write this polynomial in factored form, we will try to find its roots by either synthetic division or by finding factors of the constant term that satisfy the polynomial equation.
We look for integer factors of the constant term (-24) that could be potential roots of the polynomial. To check if a number is a root, we can use the Rational Root theorem which says that any rational root of the polynomial, where the coefficients are integers, is of the form p/q where p is a factor of the constant term and q is a factor of the leading coefficient.
Since the leading coefficient is 1 (implying q = 1), we only need to consider the factors of -24.
Possible factors of -24 include ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24. We can test these values to see which ones are roots.
Testing each one, we might find that:
f(1) = 1 - 9 + 26 - 24 = -6 (not a root)
f(2) = 8 - 36 + 52 - 24 = 0 (a root)
...
So, x = 2 is a root of the polynomial. We can now use synthetic division or long division to divide the polynomial by (x - 2) to find the other factors.
Performing synthetic division with root 2:
_______________
2 | 1 -9 26 -24
| 2 -14 24
|________________
1 -7 12 0
The quotient from the division is x^2 - 7x + 12 which can be factored further. Looking for two numbers that multiply to 12 and add up to -7, we find -3 and -4.
So, the quotient x^2 - 7x + 12 factors to (x - 3)(x - 4).
The full factored form of f(x) is:
f(x) = (x - 2)(x - 3)(x - 4)
And this is the polynomial in factored form.
