Answer :
To solve the equation [tex]\(6x^3 - 48x^4 + 90x^3 = 0\)[/tex], we want to find the values of [tex]\(x\)[/tex] that satisfy this equation. Here are the steps to reach the solution:
1. Combine Like Terms:
Begin by combining any like terms in the polynomial:
[tex]\[
6x^3 + 90x^3 = 96x^3
\][/tex]
So the equation becomes:
[tex]\[
96x^3 - 48x^4 = 0
\][/tex]
2. Factor Out the Greatest Common Factor:
Notice that both terms in the equation share a common factor. We can factor out the greatest common factor, which is [tex]\( x^3 \)[/tex]:
[tex]\[
x^3(96 - 48x) = 0
\][/tex]
3. Apply the Zero Product Property:
According to the zero product property, if a product of factors equals zero, then at least one of the factors must be zero. Therefore:
[tex]\[
x^3 = 0 \quad \text{or} \quad 96 - 48x = 0
\][/tex]
4. Solve Each Equation:
- For [tex]\(x^3 = 0\)[/tex]:
[tex]\[
x = 0
\][/tex]
- For [tex]\(96 - 48x = 0\)[/tex]:
First, solve for [tex]\(x\)[/tex] by isolating it:
[tex]\[
96 = 48x
\][/tex]
Divide both sides by 48:
[tex]\[
x = \frac{96}{48} = 2
\][/tex]
5. List All Solutions:
The solutions to the equation [tex]\(6x^3 - 48x^4 + 90x^3 = 0\)[/tex] are:
[tex]\[
x = 0 \quad \text{and} \quad x = 2
\][/tex]
These are the values of [tex]\(x\)[/tex] that satisfy the given polynomial equation.
1. Combine Like Terms:
Begin by combining any like terms in the polynomial:
[tex]\[
6x^3 + 90x^3 = 96x^3
\][/tex]
So the equation becomes:
[tex]\[
96x^3 - 48x^4 = 0
\][/tex]
2. Factor Out the Greatest Common Factor:
Notice that both terms in the equation share a common factor. We can factor out the greatest common factor, which is [tex]\( x^3 \)[/tex]:
[tex]\[
x^3(96 - 48x) = 0
\][/tex]
3. Apply the Zero Product Property:
According to the zero product property, if a product of factors equals zero, then at least one of the factors must be zero. Therefore:
[tex]\[
x^3 = 0 \quad \text{or} \quad 96 - 48x = 0
\][/tex]
4. Solve Each Equation:
- For [tex]\(x^3 = 0\)[/tex]:
[tex]\[
x = 0
\][/tex]
- For [tex]\(96 - 48x = 0\)[/tex]:
First, solve for [tex]\(x\)[/tex] by isolating it:
[tex]\[
96 = 48x
\][/tex]
Divide both sides by 48:
[tex]\[
x = \frac{96}{48} = 2
\][/tex]
5. List All Solutions:
The solutions to the equation [tex]\(6x^3 - 48x^4 + 90x^3 = 0\)[/tex] are:
[tex]\[
x = 0 \quad \text{and} \quad x = 2
\][/tex]
These are the values of [tex]\(x\)[/tex] that satisfy the given polynomial equation.
