factor this polynomial function completely
f(x)=12x^4+10x^3-214x^2+198x+90
I found all the possible rational zeros but just cant seem to factor it all
If you found the rational zeros then it is easy to find the factors.
e.g. suppose one of the values is x = -2/3
then the corresponding facor is (3x+2)
so just use your zeros and set up the factors from those following the pattern from above.
To factor the polynomial function f(x) = 12x^4 + 10x^3 - 214x^2 + 198x + 90 completely, we can follow the steps below:
Step 1: List all the possible rational zeros using the Rational Root Theorem. The rational zeros are the factors of the constant term (90) divided by the factors of the leading coefficient (12). The factors of 90 are ±1, ±2, ±3, ±5, ±6, ±9, ±10, ±15, ±18, ±30, ±45, ±90, and the factors of 12 are ±1, ±2, ±3, ±4, ±6, ±12.
Step 2: Use synthetic division to test the potential zeros or use long division to find a root. Start with one of the potential zeros and divide the polynomial by (x - r), where r is the potential zero. If the result is zero, then (x - r) is a factor of the polynomial.
Let's try dividing f(x) by (x - 1) using synthetic division:
1 | 12 10 -214 198 90
| 12 22 -192 6
------------------------
12 22 -192 6 96
The result is not zero since there is a remainder of 96.
Now let's try another potential zero. Let's try -1:
-1 | 12 10 -214 198 90
| -12 2 212 -14
------------------------
12 -2 -12 16 76
Again, the result is not zero, so -1 is not a factor.
We need to continue testing potential zeros until we find one that results in a remainder of zero. Once we find a zero, we can divide the polynomial by (x - r) in order to reduce the polynomial degree by 1.
After testing the potential zeros, we find that the polynomial does not have a rational zero, which means it cannot be factored further using rational numbers.
Alternatively, you can use a graphing calculator or software to find the zeros of the polynomial, which may be irrational or complex. Once you have the zeros, you can determine the factors of the polynomial by expressing it as a product of linear factors and quadratic factors.