Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Dec 08, 2023

Solve the polynomial equation $9y^3 - 16y = 0$ by grouping and factoring. Enter the exact solutions.

STEP 1

Assumptions
1. We are given a polynomial equation of the form $9y^3 - 16y = 0$ .
2. We will solve this equation by grouping and factoring.

STEP 2

First, we identify the common factor in each term of the polynomial.
$9y^3 - 16y = y(9y^2 - 16)$

STEP 3

We recognize that $9y^2 - 16$ is a difference of squares, which can be factored further.
$9y^2 - 16 = (3y)^2 - 4^2$

STEP 4

Apply the difference of squares factoring formula, which is $a^2 - b^2 = (a + b)(a - b)$ .
$9y^2 - 16 = (3y + 4)(3y - 4)$

STEP 5

Now we have factored the original polynomial as follows:
$y(9y^2 - 16) = y(3y + 4)(3y - 4)$

STEP 6

Set each factor equal to zero to find the solutions for $y$ .
$y = 0, \quad 3y + 4 = 0, \quad 3y - 4 = 0$

STEP 7

Solve each equation for $y$ .
For $y = 0$ :
$y = 0$
For $3y + 4 = 0$ :
$3y = -4$ $y = -\frac{4}{3}$
For $3y - 4 = 0$ :
$3y = 4$ $y = \frac{4}{3}$

STEP 8

Combine all solutions.
The solutions to the polynomial equation $9y^3 - 16y = 0$ are:
$y = 0, \quad y = -\frac{4}{3}, \quad y = \frac{4}{3}$

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