Select The Correct Answer From Each Drop-down Menu.Consider This Polynomial Equation: 6 ( X − 3 ) ( X 2 + 4 ) ( X + 1 ) = 0 6(x-3)(x^2+4)(x+1) = 0 6 ( X − 3 ) ( X 2 + 4 ) ( X + 1 ) = 0 Use The Equation To Complete This Statement.The Equation Has $\square$ Solutions. Its Real Solutions Are $x =

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Understanding Polynomial Equations

A polynomial equation is an equation in which the highest power of the variable (in this case, x) is a non-negative integer. The given equation is a polynomial equation of degree 4, as it has a term with x raised to the power of 4. The equation is given as:

6(x−3)(x2+4)(x+1)=06(x-3)(x^2+4)(x+1) = 0

Identifying the Solutions

To find the solutions of the equation, we need to set each factor equal to zero and solve for x. The equation has four factors:

  • x−3=0x-3 = 0
  • x2+4=0x^2+4 = 0
  • x+1=0x+1 = 0

Solving the First Factor

The first factor is x−3=0x-3 = 0. To solve for x, we add 3 to both sides of the equation:

x−3+3=0+3x-3+3 = 0+3

x=3x = 3

Solving the Second Factor

The second factor is x2+4=0x^2+4 = 0. To solve for x, we need to isolate the x term. However, this equation has no real solutions, as the square of any real number is non-negative, and adding 4 to it will always result in a positive number. Therefore, there are no real solutions for this factor.

Solving the Third Factor

The third factor is x+1=0x+1 = 0. To solve for x, we subtract 1 from both sides of the equation:

x+1−1=0−1x+1-1 = 0-1

x=−1x = -1

Conclusion

The equation has two real solutions: x = 3 and x = -1. The equation has two complex solutions, which are the square roots of -4. Therefore, the equation has a total of four solutions.

Answer Key

  • The equation has 4\boxed{4} solutions.
  • Its real solutions are x=3,−1x = \boxed{3, -1}.

Discussion

This problem requires the student to understand the concept of polynomial equations and how to solve them. The student needs to identify the factors of the equation, set each factor equal to zero, and solve for x. The student also needs to understand that the equation has two real solutions and two complex solutions.

Tips and Tricks

  • When solving polynomial equations, it's essential to identify the factors and set each factor equal to zero.
  • The student should use the quadratic formula to solve quadratic equations.
  • The student should be able to identify the real and complex solutions of the equation.

Real-World Applications

Polynomial equations have many real-world applications, such as:

  • Modeling population growth
  • Modeling the motion of objects
  • Solving optimization problems

Conclusion

In conclusion, solving polynomial equations requires the student to understand the concept of polynomial equations and how to solve them. The student needs to identify the factors, set each factor equal to zero, and solve for x. The student also needs to understand that the equation has two real solutions and two complex solutions.

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Q: What is a polynomial equation?

A: A polynomial equation is an equation in which the highest power of the variable (in this case, x) is a non-negative integer. The given equation is a polynomial equation of degree 4, as it has a term with x raised to the power of 4.

Q: How do I identify the solutions of a polynomial equation?

A: To find the solutions of the equation, we need to set each factor equal to zero and solve for x. The equation has four factors:

  • x−3=0x-3 = 0
  • x2+4=0x^2+4 = 0
  • x+1=0x+1 = 0

Q: What is the difference between real and complex solutions?

A: Real solutions are solutions that can be expressed as a rational number, while complex solutions are solutions that cannot be expressed as a rational number. In this case, the equation has two real solutions (x = 3 and x = -1) and two complex solutions.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, we can use the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2-4ac}}{2a}

where a, b, and c are the coefficients of the quadratic equation.

Q: What is the significance of the degree of a polynomial equation?

A: The degree of a polynomial equation is the highest power of the variable (in this case, x). The degree of the given equation is 4, which means it has a term with x raised to the power of 4.

Q: Can I use a calculator to solve polynomial equations?

A: Yes, you can use a calculator to solve polynomial equations. However, it's essential to understand the concept of polynomial equations and how to solve them, as calculators can only provide the solutions and not the steps to get there.

Q: How do I determine the number of solutions of a polynomial equation?

A: To determine the number of solutions of a polynomial equation, we need to identify the factors and set each factor equal to zero. The number of solutions is equal to the number of factors.

Q: Can I use polynomial equations to model real-world problems?

A: Yes, polynomial equations can be used to model real-world problems, such as population growth, motion of objects, and optimization problems.

Q: What are some common mistakes to avoid when solving polynomial equations?

A: Some common mistakes to avoid when solving polynomial equations include:

  • Not identifying the factors correctly
  • Not setting each factor equal to zero
  • Not solving for x correctly
  • Not checking for complex solutions

Q: How do I check if a solution is real or complex?

A: To check if a solution is real or complex, we need to express the solution in the form of a + bi, where a and b are real numbers and i is the imaginary unit. If the solution can be expressed in this form, it is a complex solution. Otherwise, it is a real solution.

Q: Can I use polynomial equations to solve systems of equations?

A: Yes, polynomial equations can be used to solve systems of equations. However, it's essential to understand the concept of systems of equations and how to solve them using polynomial equations.

Q: What are some real-world applications of polynomial equations?

A: Some real-world applications of polynomial equations include:

  • Modeling population growth
  • Modeling the motion of objects
  • Solving optimization problems
  • Solving systems of equations

Q: How do I determine the degree of a polynomial equation?

A: To determine the degree of a polynomial equation, we need to identify the highest power of the variable (in this case, x). The degree of the given equation is 4, which means it has a term with x raised to the power of 4.