Question

x^3-8x^2+1=56 <--Find the solutions using the Fundamental Theorem of Algebra, the Rational Root Theorem, Descartes' Rule of Signs, and the Factor Theorem.

Answer 1

first subtract 56 to get
\[x^3-8x^2-55=0\]

Answer 2

now the possible rational zeros are divisors of 55, namely
\[\pm1, \pm5,\pm11,\pm55\]

Answer 3

1 and -1 clearly don't work

Answer 4

hold the phone are you sure this is right? because i don't think this has any rational zeros. just one irrational one

Answer 5

I just have to find the zeros using the things mentioned in the question

Answer 6

no it does not
you have no hope here, there is no rational zero.

Answer 7

are you sure it is
\[x^3-8x^2+1=56\]?

Answer 8

Procedure:

Measure and record the length, width and height of the rectangular box you have chosen in inches. Round to the nearest whole number.

Apply the formula of a rectangular box (V = lwh) to find the volume of the object.

Now suppose you knew the volume of this object and the relation of the length to the width and height, but did not know the length. Rewriting the equation with one variable would result in a polynomial equation that you could solve to find the length.

Rewrite the formula using the variable x for the length. Substitute the value of the volume found in step 2 for V and express the width and height of the object in terms of x plus or minus a constant. For example, if the height measurement is 4 inches longer than the length, then the expression for the height will be (x + 4).

Simplify the equation and write it in standard form.
Find the solutions to this equation algebraically using the Fundamental Theorem of Algebra, the Rational Root Theorem, Descartes' Rule of Signs, and the Factor Theorem.

(this is what i had to do, and my length was:7inches, my width was 4 inches, and my height was 2 inches...did i do something wrong?

Answer 9

ok so volume is 56

Answer 10

x = 7

Answer 11

w = x-3
h = x-5

Answer 12

so i guess you are supposed to write
\[x(x-3)(x-5)=56\] and solve for x

Answer 13

but then how would i simplify it and get it into standard form?

Answer 14

\[x^3-8x^2+15x=56\]

Answer 15

or rather
\[x^3-8x^2+15x-56=0\]

Answer 16

now you are just supposed to work backwards.

Answer 17

ohh ok

Answer 18

you know that the rational zeros are divisors of 56, but of course you already know the answer since those are the numbers you started with

Answer 19

zeros are 7, 4 and 2, but if you want to impress your teacher write down the possible rational zeros and try them

Answer 20

i can never remember descates rule of sign, but once you have a zero (say 2) you can use the factor theorem to write
\[x^3-8x^2+15x-56=(x-2)q(x)\]

Answer 21

when you are done you will have
\[x^3-8x^2+15x-56=(x-2)(x-4)(x-7)\]