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 divide the function f(x)=x³ + 14x² + 57x + 72 by the width of the dvd stand, x + 3 units. this gives the polynomial x² + 11x + 24. 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 (simplify your answers. use a comma to separate answers as needed.)

Explanation:

Step1: Divide the volume polynomial

We divide the volume function $f(x)=x^{3}+14x^{2}+57x + 72$ by the width $x + 3$ using polynomial long - division or synthetic division. By synthetic division:

• The coefficients of the dividend are $1,14,57,72$.
• The zero of the divisor $x+3$ is $x=-3$.
• Performing 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 polynomial

We 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 $8$ since $3\times8 = 24$ and $3 + 8=11$.
So, $x^{2}+11x + 24=(x + 3)(x+8)$. But we already used the factor $x + 3$ as the width, so the other two possible non - width factors for length and height are $x + 8$ and $x+3$ (we can consider different assignments for length and height). Since we are looking for non - repeated factors other than the given width factor for the remaining dimensions, the possible non - width dimensions for length and height are $x + 8$ units.

Answer:

$x + 8$