Question

the function, ( p(x) = x^3 - 4x^2 - 20x + 48 ) has a factor of ( x - 2 ). which is ( p(x) ) written as a product of linear factors?
a ( p(x) = (x - 2)(x - 4)(x + 6) )
b ( p(x) = (x - 2)(x + 4)(x - 6) )
c ( p(x) = (x - 2)(x + 2)(x + 6) )
d ( p(x) = (x - 2)(x - 3)(x + 4) )

Explanation:

Step1: Perform polynomial division or use factoring

Since \( p(x) = x^3 - 4x^2 - 20x + 48 \) has a factor of \( (x - 2) \), we can use polynomial long division or synthetic division to divide \( p(x) \) by \( (x - 2) \). Let's use synthetic division.
The root for \( (x - 2) \) is \( x = 2 \).
Set up synthetic division:
\[

$$\begin{array}{r|rrrr} 2 & 1 & -4 & -20 & 48 \\ & & 2 & -4 & -48 \\ \hline & 1 & -2 & -24 & 0 \\ \end{array}$$

\]
So, the quotient polynomial is \( x^2 - 2x - 24 \).

Step2: Factor the quadratic polynomial

Now, factor \( x^2 - 2x - 24 \). We need two numbers that multiply to \( -24 \) and add to \( -2 \). The numbers are \( -6 \) and \( 4 \) (since \( -6 \times 4 = -24 \) and \( -6 + 4 = -2 \)).
So, \( x^2 - 2x - 24 = (x - 6)(x + 4) \).

Step3: Write \( p(x) \) as a product of linear factors

Since \( p(x) = (x - 2)(x^2 - 2x - 24) \), substituting the factored quadratic, we get \( p(x) = (x - 2)(x - 6)(x + 4) \), which can be rewritten as \( p(x) = (x - 2)(x + 4)(x - 6) \) (which is option B).

Answer:

B. \( p(x) = (x - 2)(x + 4)(x - 6) \)