Question

The polynomial 2x^3+9x^2+4x-15 demonstrates the volume of a packaging box at a shipping company. The box is (x-1) feet from lid to base and is 13 feet long. Find the linear factors of the polynomial. Which linear factor denotes the length of 13 feet? In feet, what are the length, width, and height of the box? Calculate the volume of the box and solve for x.

Answer 1

First step: Divide 2x^3+9x^2+4x-15 by (x-1). Use synthetic division.

Answer 2

2x^2+11x+15?

Answer 3

Okay. Second step: Factor the quadratic expression: 2x^2+11x+15

Answer 4

(x+3)(2x+5)

Answer 5

Yes!
Volume of a box = Length * Width * Height

Volume = 2x^3+9x^2+4x-15 = (x-1) * (x+3) * (2x+5)

We have to figure out which factor corresponds to the length, which factor corresponds to the width and which factor corresponds to the height.

"The box is (x-1) feet from lid to base"
This refers to the height of the box.
So height = (x-1).

In using the terms length and width, we usually refer to the longer dimension as the length and the shorter dimension as the width. So between the two remaining factors, (2x+5) and (x+3) which one would represent the length?

Answer 6

(2x+5) would be the length right?

Answer 7

Yes!
" ... box .... is 13 feet long."
Therefore, length 2x + 5 = 13. Solve for x.
Then put that x value and calculate the following:
Length = 2x + 5 = ? feet
Width = x + 3 = ? feet
Height = x - 1 = ? feet
Volume = Length * Width * height = ? feet^3

Answer 8

x=3 so 13 feet long, 6 feet wide, and 2 feet high. So the volume is 156feet^3.

Answer 9

2x + 5 = 13
subtract 5
2x = 8
x = 4

Answer 10

13 feet long, 8 feet wide, 3 feet high, 312 feet^3?

Answer 11

x = 4
Length = 13 feet (given)
Width = x + 3 = 4 + 3 = 7 feet (not 8 ft.)
Height = x - 1 = 4 - 1 = 3 feet

Volume = 13 * 7 * 3 = 273 ft^3

Just to make sure, put x = 4 in Volume = 2x^3+9x^2+4x-15
= 2(4)^3 + 9(4)^2 + 4(4) - 15
= 128 + 144 + 16 - 15
= 273

Answer 12

And I also need to find the relationship between the graph of the polynomial and the linear factors but I cant seem to find any relation.

Answer 13

Here is the graph of the polynomial 2x^3+9x^2+4x-15:

http://prntscr.com/42bsbo

Answer 14

Is the relation that the x intercept is at 3 which is (x+3), 1 which is (x-1) and 2.5 (2x+5)?

Answer 15

If we know the three roots of a polynomial, p, q, r, then the polynomial is:
A(x-p)(x-q)(x-r) (where A is a constant).
From the graph we can see the roots are: -3, -2.5 or -5/2, +1
Therefore, the polynomial is: A(x+3)(x+5/2)(x-1) = A(x+3)( (2x+5)/2 )(x-1)
If A = 2, then the polynomial is (x+3)(2x+5)(x-1)

Answer 16

So we started with the three roots from the graph, constructed a polynomial, then chose a suitable value for the constant A and we arrived at the given polynomial which we factored earlier and showed it is (2x+5)(x+3)(x-1).

Answer 17

If the x-intercept of a polynomial is p, then (x-p) is a factor of the polynomial.