Answer :
Let's address the task of solving the provided polynomial equations and understand their solutions step-by-step.
We have two polynomial equations:
1. [tex]\( y = 3x^3 - 7x^2 + 5 \)[/tex]
2. [tex]\( y = 7x^4 + 2x \)[/tex]
We need to solve the following equations:
1. [tex]\( 7x^4 + 2x = 0 \)[/tex]
2. [tex]\( 7x^4 + 3x^3 - 7x^2 + 2x + 5 = 0 \)[/tex]
3. [tex]\( 3x^3 - 7x^2 + 5 = 0 \)[/tex]
4. [tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex]
### Solving Equation 1:
[tex]\[ 7x^4 + 2x = 0 \][/tex]
To solve this, factor out [tex]\( x \)[/tex]:
[tex]\[ x(7x^3 + 2) = 0 \][/tex]
This means:
- [tex]\( x = 0 \)[/tex] or
- [tex]\( 7x^3 + 2 = 0 \)[/tex]
For [tex]\( 7x^3 + 2 = 0 \)[/tex], solve for [tex]\( x \)[/tex]:
[tex]\[ 7x^3 = -2 \][/tex]
[tex]\[ x^3 = -\frac{2}{7} \][/tex]
[tex]\[ x = \left(-\frac{2}{7}\right)^{1/3} \][/tex]
The solutions for this equation are:
- [tex]\( x = 0 \)[/tex]
- [tex]\( x = \left(-\frac{2}{7}\right)^{1/3} \)[/tex] and two other complex roots as part of the complete solution.
### Solving Equation 2:
[tex]\[ 7x^4 + 3x^3 - 7x^2 + 2x + 5 = 0 \][/tex]
This is a quartic (degree 4) equation, which can be quite complex to solve analytically without specific techniques, such as using technology for numerical approximations or methods like factoring, using the Rational Root Theorem, or synthetic division, which may not always work nicely without more information or simplification.
The solutions to this equation are complex and include real and imaginary parts.
### Solving Equation 3:
[tex]\[ 3x^3 - 7x^2 + 5 = 0 \][/tex]
This is a cubic equation. Solving it typically involves trying possible rational roots using the Rational Root Theorem or finding numerical solutions. Again, without more straightforward factoring options, technology can quickly find approximate or exact solutions.
The solutions to this equation also include complex numbers.
### Solving Equation 4:
This involves setting the two original equations equal:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
Rearranging gives Equation 2, which is already discussed. The solutions for it would be exactly the same.
### Conclusion:
These polynomial equations have solutions that include both real and complex numbers. The solutions involve roots that may not always be expressed simply without complex radicals. All these solutions can be considered accurate.
We have two polynomial equations:
1. [tex]\( y = 3x^3 - 7x^2 + 5 \)[/tex]
2. [tex]\( y = 7x^4 + 2x \)[/tex]
We need to solve the following equations:
1. [tex]\( 7x^4 + 2x = 0 \)[/tex]
2. [tex]\( 7x^4 + 3x^3 - 7x^2 + 2x + 5 = 0 \)[/tex]
3. [tex]\( 3x^3 - 7x^2 + 5 = 0 \)[/tex]
4. [tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex]
### Solving Equation 1:
[tex]\[ 7x^4 + 2x = 0 \][/tex]
To solve this, factor out [tex]\( x \)[/tex]:
[tex]\[ x(7x^3 + 2) = 0 \][/tex]
This means:
- [tex]\( x = 0 \)[/tex] or
- [tex]\( 7x^3 + 2 = 0 \)[/tex]
For [tex]\( 7x^3 + 2 = 0 \)[/tex], solve for [tex]\( x \)[/tex]:
[tex]\[ 7x^3 = -2 \][/tex]
[tex]\[ x^3 = -\frac{2}{7} \][/tex]
[tex]\[ x = \left(-\frac{2}{7}\right)^{1/3} \][/tex]
The solutions for this equation are:
- [tex]\( x = 0 \)[/tex]
- [tex]\( x = \left(-\frac{2}{7}\right)^{1/3} \)[/tex] and two other complex roots as part of the complete solution.
### Solving Equation 2:
[tex]\[ 7x^4 + 3x^3 - 7x^2 + 2x + 5 = 0 \][/tex]
This is a quartic (degree 4) equation, which can be quite complex to solve analytically without specific techniques, such as using technology for numerical approximations or methods like factoring, using the Rational Root Theorem, or synthetic division, which may not always work nicely without more information or simplification.
The solutions to this equation are complex and include real and imaginary parts.
### Solving Equation 3:
[tex]\[ 3x^3 - 7x^2 + 5 = 0 \][/tex]
This is a cubic equation. Solving it typically involves trying possible rational roots using the Rational Root Theorem or finding numerical solutions. Again, without more straightforward factoring options, technology can quickly find approximate or exact solutions.
The solutions to this equation also include complex numbers.
### Solving Equation 4:
This involves setting the two original equations equal:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
Rearranging gives Equation 2, which is already discussed. The solutions for it would be exactly the same.
### Conclusion:
These polynomial equations have solutions that include both real and complex numbers. The solutions involve roots that may not always be expressed simply without complex radicals. All these solutions can be considered accurate.
