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When the cube of a non-zero number \(y\) is subtracted from \(35,\) the result is equal to the result of dividing 216 by the cube of that number \(y .\) What is the sum of all the possible values of \(y ?\) a. \(\frac{5}{2}\) b. 5 c. 6 d. 10 e. 12

Short Answer

Expert verified
The sum of all possible values of \(y\) is 5.

Step by step solution

01

Set up the equation

Translate the problem statement into a mathematical equation. Given: When the cube of a non-zero number \(y\) is subtracted from \(35\), the result is equal to the result of dividing \(216\) by the cube of that number \(y\). Thus, the equation is: 35 - y^3 = \frac{216}{y^3}.
02

Eliminate the fraction

Multiply both sides of the equation by \(y^3\) to eliminate the fraction: y^3 (35 - y^3) = 216.
03

Simplify the equation

Distribute \(y^3\) on the left-hand side and simplify: 35y^3 - y^6 = 216 Rearrange to form a standard polynomial equation: y^6 - 35y^3 + 216 = 0.
04

Solve for \(z = y^3\)

Introduce a substitution \(z = y^3\) to turn the equation into a quadratic form: z^2 - 35z + 216 = 0.
05

Solve the quadratic equation

Solve for \(z\) using the quadratic formula: \[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 1\), \(b = -35\), and \(c = 216\). \[ z = \frac{35 \pm \sqrt{35^2 - 4 \cdot 1 \cdot 216}}{2 \cdot 1} \] \[ z = \frac{35 \pm \sqrt{1225 - 864}}{2} \] \[ z = \frac{35 \pm \sqrt{361}}{2} \] \[ z = \frac{35 \pm 19}{2} \] Thus, the solutions are \(z = 27\) and \(z = 8\).
06

Determine the values of \(y\)

Since \(z = y^3\), find the corresponding values of \(y\): If \(z = 27\), then \(y^3 = 27\) implies \(y = 3\). If \(z = 8\), then \(y^3 = 8\) implies \(y = 2\).
07

Sum all possible values of \(y\)

Finally, find the sum of all possible values of \(y\): \(\text{Sum} = 3 + 2 = 5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cube of a Number
The cube of a number refers to raising a number to the third power. For a given number \(y\), its cube is represented as \(y^3\). This means multiplying the number by itself twice more: \(y \times y \times y = y^3\).

For example:
  • If \(y = 2\), then \(2^3 = 2 \times 2 \times 2 = 8\).
  • If \(y = 3\), then \(3^3 = 3 \times 3 \times 3 = 27\).

In the given problem, you need to solve an equation where a cube of a number plays a key role in both subtracting from 35 and dividing a constant.
Polynomial Equation
A polynomial equation involves variables raised to different powers with constant coefficients. They can range from simple (linear) to complex (higher-degree polynomials). The equation from our problem is an example of a polynomial equation:

When we arrived at
\[ y^6 - 35y^3 + 216 = 0 \], we had a polynomial equation of degree 6. These types of equations often have multiple solutions, as each root corresponds to a potential value of the variable.
Quadratic Formula
The quadratic formula is used to solve quadratic equations of the form \(ax^2 + bx + c = 0\). The solutions for \(x\) can be found using the formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In the problem, substituting \(z = y^3\) converted our original polynomial to a quadratic form \(z^2 - 35z + 216 = 0\).

Here,
  • \(a = 1\)
  • \(b = -35\)
  • \(c = 216\)
. Using the quadratic formula, we found the solutions for \(z\) which helped us find the values for \(y\).
Substitution Method
The substitution method involves replacing one variable with another expression to simplify an equation. This is particularly helpful with polynomial equations that can be transformed into quadratic equations.

In this exercise, we introduced a substitution \(z = y^3\). This substitution recast our original sixth-degree polynomial into the quadratic equation \(z^2 - 35z + 216 = 0\).

Solving the quadratic equation for \(z\) then allowed us to back-substitute to find the values of \(y\). This technique is invaluable for transforming and solving higher-degree equations in mathematics.

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Most popular questions from this chapter

If \(x>0, y>0,\) and \(\frac{7 x^{2}+72 x y+4 y^{2}}{4 x^{2}+12 x y+5 y^{2}}=4,\) what is the value of \(\frac{x+y}{y} ?\) a. \(\frac{3}{4}\) b. \(\frac{4}{3}\) c. \(\frac{10}{7}\) d. \(\frac{7}{4}\) e. \(\frac{7}{3}\)

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If \(a\) and \(b\) are integers, and \(2 a+b=17,\) then \(8 a+b\) cannot equal which of the following? a. -1 b. 33 c. 35 d. 65 e. 71

The numbers \(m, n,\) and \(T\) are all positive, and \(m>n .\) A weekend farm stand sells only peaches. On Saturday, the farm stand has \(T\) peaches to sell, at a profit of \(m\) cents each. Any peaches remaining for sale on Sunday will be marked down and sold at a profit of \((m-n)\) cents each. If all peaches available for sale on Saturday morning are sold by Sunday evening, how many peaches, in terms of \(T, m,\) and \(n,\) does the stand need to sell on Saturday in order to make the same profit on each day? a. \(\frac{T m}{m-n}\) b. \(\frac{T m}{n-m}\) c. \(\frac{m(m-n)}{T}\) d. \(\frac{T}{m-n}\) e. \(\frac{T(m-n)}{2 m-n}\)

There are 816 students in enrolled at a certain high school. Each of these students is taking at least one of the subjects economics, geography, and biology. The sum of the number of students taking exactly one of these subjects and the number of students taking all 3 of these subjects is 5 times the number of students taking exactly 2 of these subjects. The ratio of the number of students taking only the two subjects economics and geography to the number of students taking only the two subjects economics and biology to the number of students taking only the two subjects geography and biology is \(3: 6: 8 .\) How many of the students enrolled at this high school are taking only the two subjects geography and biology? a. 35 b. 42 c. 64 d. 136 e. 240

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