5z^8/3+4z^5/3-12z^2/3=0
The instructions say solve the following equation by factoring. If need be submit the answer by fraction reduced to lowest terms
factor out z^2/3:
z^2/3 (5z^2 + 4z - 12) = 0
z^2/3 (5z-6)(z+2) = 0
That should help
To solve the equation 5z^8/3 + 4z^5/3 - 12z^2/3 = 0, we can follow these steps:
Step 1: Rewrite the equation in a simpler form:
5z^(8/3) + 4z^(5/3) - 12z^(2/3) = 0
Step 2: To simplify the equation, let's introduce a substitution. Let's set y = z^(1/3). This will help us eliminate the fractional exponents.
Substituting y into the equation, we get:
5y^8 + 4y^5 - 12y^2 = 0
Step 3: Now we have a polynomial equation in terms of y. We can try factoring it to find the values of y:
5y^8 + 4y^5 - 12y^2 = 0
Factoring out the common factor of y^2, we have:
y^2(5y^6 + 4y^3 - 12) = 0
Step 4: Set each factor equal to zero and solve for y:
First factor: y^2 = 0
This implies y = 0.
Second factor: 5y^6 + 4y^3 - 12 = 0
This equation, unfortunately, cannot be easily factored further.
Step 5: Since we have set y = z^(1/3), we can substitute y back into the equation.
Substituting y = 0 into y = z^(1/3), we get:
0 = z^(1/3)
This equation tells us that z = 0.
Substituting y = z^(1/3) into the second factor, we get:
5(z^(1/3))^6 + 4(z^(1/3))^3 - 12 = 0
5z^2 + 4z^(1/3) - 12 = 0
Step 6: Now we have a quadratic equation in terms of z. We can solve it using the quadratic formula.
Using the quadratic formula, we have:
z = (-b ± √(b^2 - 4ac)) / (2a)
For our equation 5z^2 + 4z^(1/3) - 12 = 0, a = 5, b = 4, and c = -12.
Substituting these values into the quadratic formula, we get:
z = (-4 ± √(4^2 - 4(5)(-12)) / (2(5))
z = (-4 ± √(16 + 240)) / 10
z = (-4 ± √256) / 10
z = (-4 ± 16) / 10
Therefore, we have two possible solutions for z:
z = (16 - 4) / 10 = 12/10 = 1.2
z = (-4 - 16) / 10 = -20/10 = -2
So the solutions to the equation 5z^8/3 + 4z^5/3 - 12z^2/3 = 0 are z = 0, z = 1.2, and z = -2.