Question

Use synthetic division to completely factor y=x^3-3x^2-10x+24 by x – 2.A. y = (x – 2)(x – 4)(x – 3)B. y = (x – 2)(x + 4)(x + 3)C. y = (x – 2)(x + 4)(x – 3)D. y = (x – 2)(x – 4)(x + 3)

Answer 1

D. y = (x – 2)(x – 4)(x + 3)

Explanation:

We want to completely factor the polynomial.

First, given the division of polynomials below:

`(x^3-3x^2-10x+24)/(x-2)`

To use synthetic division, set the denominator equal to 0 and solve for x:

`x-2=0\implies x=2`

Next, write the result(2) outside and the coefficients of the numerator inside as shown below:

Carry down the leading coefficient as shown below:

Multiply the carry down value(1) by the number outside (2) and write the number under the next column:

Repeat until it gets to the last column:

Therefore:

`(x^(3)-3x^(2)-10x+24)/(x-2)=x^2-x-12`

We factor the resulting quadratic expression:

`\begin{gathered} x^2-x-12=x^2-4x+3x-12 \\ =x(x-4)+3(x-4) \\ =(x+3)(x-4) \end{gathered}`

So, we have:

`\begin{gathered} (x^(3)-3x^(2)-10x+24)/(x-2)=(x+3)(x-4) \\ \implies x^3-3x^2-10x+24=(x-2)\left(x-4\right)(x+3) \end{gathered}`

Thus:

`y=(x-2)\left(x-4\right)(x+3)`

Option D is correct.