Question

dajuan wants to determine the length and height of his dvd stand. the function f(x)=x³ + 14x²+57x + 72 represents the volume of the dvd stand, where the width is x + 3 units. what are possible dimensions for the length and height of the dvd stand? explain.
to find the possible dimensions for the length and height of the dvd stand, first blank the function f(x)=x³ + 14x²+57x + 72 by the width of the dvd stand, x + 3 units. this gives the polynomial blank. the factors of this polynomial are the possible dimensions for the length and height. so, the possible dimensions for the length and height of the dvd stand are blank units. (simplify your answers. use a comma to separate answers as needed.)

Explanation:

Step1: Divide the polynomial

We divide the polynomial $f(x)=x^{3}+14x^{2}+57x + 72$ by $x + 3$ using polynomial long - division or synthetic division. Using synthetic division:
The coefficients of $f(x)$ are $1,14,57,72$. We use $- 3$ (since $x+3=x-(-3)$) for synthetic division.
Bring down the first coefficient $1$:
Multiply $-3\times1=-3$, add to the second coefficient: $14+( - 3)=11$.
Multiply $-3\times11=-33$, add to the third coefficient: $57+( - 33)=24$.
Multiply $-3\times24=-72$, add to the fourth coefficient: $72+( - 72)=0$.
The quotient is $x^{2}+11x + 24$.

Step2: Factor the quotient

Factor the quadratic polynomial $x^{2}+11x + 24$.
We need to find two numbers that multiply to $24$ and add up to $11$. The numbers are $3$ and $8$.
So, $x^{2}+11x + 24=(x + 3)(x+8)$.

Answer:

$x + 8,x + 3$