The factors of x^3-4x^2-x+4 are?
Factor by grouping.
Factor x^2 out of the first 2 terms, and -1 out of the second 2 terms.
= x^2(x-4)-1(x-4)
Now we have the second factor of (x^2-1)
= (x^2-1)(x-4)
Factor x^2-1.
= (x-1)(x+1)(x-4)
how did you find (x^2-1)?
oh, i understand. Thank you.
To find the factors of a polynomial, we can use the factor theorem and synthetic division. We need to test different values for x to see if they are roots of the polynomial.
The factors of a polynomial are the values of x that make the polynomial equal to zero.
First, let's list all possible factors of the constant term (in this case, 4). The factors of 4 are ±1, ±2, and ±4.
Next, we will test each of these values by performing synthetic division to determine if they are roots of the polynomial.
Synthetic division is a process that allows us to divide a polynomial by a factor (x - a) and find the quotient and remainder.
Using synthetic division, we test each possible factor by dividing the polynomial by (x - a), where a is the value we are testing.
For example, to test if x - 1 is a factor, we use synthetic division as follows:
1 | 1 -4 -1 4
1 -3 -4
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1 -3 -4 0
Since the remainder is 0, we can conclude that x - 1 is a factor of the polynomial.
Now, we repeat the process for the resulting quadratic polynomial, which is x^2 - 3x - 4, to find its factors.
Again, we list all possible factors of the constant term (-4), which are ±1, ±2, and ±4.
We perform synthetic division using these values to test for factors.
For example, testing x - 1 as a factor:
1 | 1 -3 -4
1 -2
________________
1 -2 -6
The remainder is not 0, so x - 1 is not a factor of x^2 - 3x - 4.
By repeating the synthetic division process for each possible factor, we find that the factors of x^3 - 4x^2 - x + 4 are (x - 1) and (x^2 - 2x - 4).