Final answer:
To find the quotient polynomial q(x) such that p(x) = (q(x))(x - 1) for the polynomial p(x) = x^3 - 3x^2 + 4x - 12, we can use synthetic division by dividing the polynomial by the binomial factor x - 1, which yields the coefficients of q(x).
Step-by-step explanation:
The student's question involves dividing a polynomial by a binomial, where p(x) = x^3 - 3x^2 + 4x - 12 is the polynomial and it is given that p(x) = (q(x))(x - 1). The question seeks to find the quotient polynomial q(x). To solve this, we can perform polynomial long division or synthetic division, since x - 1 is a linear factor.
We will use synthetic division because it is simpler with linear factors. First, we evaluate if x = 1 is a root of the polynomial by plugging it into the polynomial. If p(1) equals zero, then (x - 1) is indeed a factor. After confirming this, we can divide the polynomial by x - 1 using synthetic division.
The process will look something like this:
- Put the coefficients of p(x) in a row: 1, -3, 4, -12.
- Write down the zero of the binomial x - 1, which is x = 1, on the left side of the coefficients.
- Bring down the first coefficient (1).
- Multiply the zero (1) by the first coefficient (1) and write the result under the second coefficient (-3).
- Add the second coefficient (-3) and the result of the multiplication (1), placing the result under the line.
- Continue this process for the rest of the coefficients.
The result of the synthetic division will give us the coefficients of q(x) which, when multiplied by (x - 1), will give us p(x).