Answer :
To solve the equation [tex]\( 12x^6 + 10x^5 + 48x^4 + 40x^3 = 0 \)[/tex] algebraically and find all values of [tex]\( x \)[/tex], follow these steps:
1. Factor out the greatest common factor:
Notice that each term on the left side has a common factor of [tex]\( 2x^3 \)[/tex].
[tex]\[
12x^6 + 10x^5 + 48x^4 + 40x^3 = 2x^3(6x^3 + 5x^2 + 24x + 20) = 0
\][/tex]
2. Set the factors equal to zero:
For a product to be zero, at least one of the factors must be zero. Therefore, set each factor to zero:
[tex]\[
2x^3 = 0 \quad \text{or} \quad 6x^3 + 5x^2 + 24x + 20 = 0
\][/tex]
The first factor [tex]\( 2x^3 = 0 \)[/tex]:
[tex]\[
x^3 = 0 \implies x = 0
\][/tex]
3. Solve the remaining polynomial:
Now we need to solve the cubic polynomial [tex]\( 6x^3 + 5x^2 + 24x + 20 = 0 \)[/tex].
By attempting to solve this equation, we find that it can have both real and complex roots.
4. Details of the roots:
The solutions to the polynomial equation [tex]\( 6x^3 + 5x^2 + 24x + 20 = 0 \)[/tex] are:
[tex]\[
x = -\frac{5}{6}, \quad x = -2i, \quad x = 2i
\][/tex]
Therefore, the complete solutions to the equation [tex]\( 12x^6 + 10x^5 + 48x^4 + 40x^3 = 0 \)[/tex] are:
[tex]\[
\boxed{0, -\frac{5}{6}, -2i, 2i}
\][/tex]
1. Factor out the greatest common factor:
Notice that each term on the left side has a common factor of [tex]\( 2x^3 \)[/tex].
[tex]\[
12x^6 + 10x^5 + 48x^4 + 40x^3 = 2x^3(6x^3 + 5x^2 + 24x + 20) = 0
\][/tex]
2. Set the factors equal to zero:
For a product to be zero, at least one of the factors must be zero. Therefore, set each factor to zero:
[tex]\[
2x^3 = 0 \quad \text{or} \quad 6x^3 + 5x^2 + 24x + 20 = 0
\][/tex]
The first factor [tex]\( 2x^3 = 0 \)[/tex]:
[tex]\[
x^3 = 0 \implies x = 0
\][/tex]
3. Solve the remaining polynomial:
Now we need to solve the cubic polynomial [tex]\( 6x^3 + 5x^2 + 24x + 20 = 0 \)[/tex].
By attempting to solve this equation, we find that it can have both real and complex roots.
4. Details of the roots:
The solutions to the polynomial equation [tex]\( 6x^3 + 5x^2 + 24x + 20 = 0 \)[/tex] are:
[tex]\[
x = -\frac{5}{6}, \quad x = -2i, \quad x = 2i
\][/tex]
Therefore, the complete solutions to the equation [tex]\( 12x^6 + 10x^5 + 48x^4 + 40x^3 = 0 \)[/tex] are:
[tex]\[
\boxed{0, -\frac{5}{6}, -2i, 2i}
\][/tex]
