Question

A. Use synthetic division an(d)/(o)r factoring to write f(x)=2x^(3)-3x^(2)-11x+6 in completely factored form, given that (x-3) is a factor of f(x).

Answer 1

The polynomial f(x)=2x^3-3x^2-11x+6 can be factored using synthetic division with the known factor (x-3), resulting in f(x)=(x-3)(2x-1)(x+2) as the completely factored form.

Step-by-step explanation:

To find the completely factored form of the polynomial f(x)=2x^3-3x^2-11x+6, given that (x-3) is a factor, we can use synthetic division or factoring. Since we already have one factor, (x-3), we will perform synthetic division using 3 as the zero of the factor to divide the polynomial.

Performing synthetic division, we place 3 in the division box and the coefficients of )f(x beside it: 2, -3, -11, 6. After carrying out the division, we get a quotient of 2x^2+3x-2 and a remainder of 0, confirming that (x-3) is indeed a factor. Now, we need to factor the quadratic 2x^2+3x-2.

Factoring the quadratic, we get (2x-1)(x+2). Therefore, the completely factored form of the polynomial is: f(x)=(x-3)(2x-1)(x+2). Finally, we eliminate terms wherever possible to simplify the algebra and check the answer to see if it is reasonable.

Answer 2

The complete factored form is
`(x-3)(2 x+3)(3 x-2)`

Find one factor using synthetic division:

We can perform synthetic division to find the quotient when
`2 x^3-3 x^2-11 x+6` is divided by (x−3).

3 | 2 -3 -11 6

|-------

| 6 3 -6

| - 3 6

| ------

0 0 0 0

The quotient is
`6 x^2+3 x-6`, and the remainder is 0. Therefore, we can write:

`2 x^3-3 x^2-11 x+6=(x-3)\left(6 x^2+3 x-6\right)`

Find the remaining factors:

Now, we need to factor the remaining quadratic
`6 x^2+3 x-6`. We can use factoring by grouping:

`6 x^2+3 x-6=(2 x+3)(3 x-2)`

Combine the factors:

Combining the factors from steps 1 and 2, we get the complete factored form:

`(x-3)(2 x+3)(3 x-2)`