how do you factor the polynomial 3x^4-4x^3+4x^2-4x+1?
i got (x-1)(3x-1)(3x^2+3)
but the answer is (x-1)(3x-1)(x+i)(x-i)
You should have realized your answer was wrong by looking at the constants of your three factors.
Multiplying them would have given you a last term of +3 instead of the needed +1.
You were close, actually you are off by only a constant factor of 3
Look at your last factor of (3x^2+3)
=3(x^2+1)
=3(x+i)(x-i)
Compare that with the actual answer given.
Looks like you just made an arthmetic error somewhere in your division.
3x^2+3 = 3 x(x^2 + 1) = 3(x+i)(x-i)
You only got an overall factor of 3 wrong.
where does the "3" go then?
3x^2+3 = 3 x(x^2 + 1) =
"3"(x+i)(x-i)
because the answer is:
(x-1)(3x-1)(x+i)(x-i)
To factor the polynomial 3x^4 - 4x^3 + 4x^2 - 4x + 1, let's follow these steps:
Step 1: Check for a common factor.
In this case, there is no common factor among the terms, so we can proceed to the next step.
Step 2: Look for a factor of the form (x - a).
We can try substituting different values into the polynomial to find a factor. Let's start with a = 1.
When we substitute x = 1 into the polynomial, we get:
3(1)^4 - 4(1)^3 + 4(1)^2 - 4(1) + 1 = 3 - 4 + 4 - 4 + 1 = 0
Since we obtained zero, (x - 1) is a factor of the polynomial.
Step 3: Use polynomial long division or synthetic division.
To proceed with factoring, we need to divide the polynomial by (x - 1). If we perform polynomial long division or synthetic division, we get:
3x^3 + (-1x^2) + 3x + (-1)
-------------------------------------------
x - 1 | 3x^4 - 4x^3 + 4x^2 - 4x + 1
This results in a quotient of 3x^3 - x^2 + 3x - 1.
Step 4: Continue factoring the quotient.
Now, we focus on factoring the quotient 3x^3 - x^2 + 3x - 1. We can repeat the process by trying a = 1 again. However, this time it does not yield zero.
At this point, we can notice that the quadratic equation x^2 + 1 = 0 does not have any real roots. In this case, we need to consider complex roots. The solutions to x^2 + 1 = 0 are x = i and x = -i, where i represents the imaginary unit (√(-1)).
Therefore, the remaining factors of the polynomial are (x + i) and (x - i).
Putting it all together, the factored form of the polynomial 3x^4 - 4x^3 + 4x^2 - 4x + 1 is:
(x - 1)(3x - 1)(x + i)(x - i)