Question

The volume v in cubic feet of a shipping box is modeled by the polynomial function V(x)=x^3-2x^2-19x+20, where X is the length of the box. Show that x+4 is a factor of v(x). Then factor v(x) completely

Answer 1

To show that x + 4 is a factor of V(x), we can use the Remainder Theorem. If V(-4) = 0, then x + 4 is indeed a factor.

Evaluate V(-4):

V(-4) = (-4)^3 - 2(-4)^2 - 19(-4) + 20
V(-4) = -64 - 2(16) + 76 + 20
V(-4) = -64 - 32 + 76 + 20
V(-4) = 0

Since V(-4) equals 0, we can conclude that x + 4 is a factor of V(x).

Now, to factor V(x) completely, we can use synthetic division or polynomial long division to divide V(x) by (x + 4):

The result of the division is the quotient 1x^2 - 6x + 7 and a remainder of 8.

So, V(x) can be factored as:

V(x) = (x + 4)(x^2 - 6x + 7)

Now, let's further factor the quadratic expression x^2 - 6x + 7:

x^2 - 6x + 7 = (x - 1)(x - 7)

Now, we have factored V(x) completely:

V(x) = (x + 4)(x - 1)(x - 7)

So, the complete factorization of V(x) is
(x + 4)(x - 1)(x - 7).