Solve For { X $}$:i) { 6x^3 + 16x^2 + 8x = 0 $}$Given { X = -\frac{1}{3} $}$.
Introduction
When solving polynomial equations, it's essential to understand the properties of the coefficients and the degree of the polynomial. In this case, we're given a cubic equation { 6x^3 + 16x^2 + 8x = 0 $}$ and we need to find the value of { x $}$ that satisfies this equation. We're also given a hint that { x = -\frac{1}{3} $}$ is a solution to the equation.
Understanding the Equation
The given equation is a cubic equation, which means it has a degree of 3. This means that the highest power of { x $}$ in the equation is 3. The equation can be written as { 6x^3 + 16x^2 + 8x = 0 $}$. We can see that the coefficients of the equation are 6, 16, and 8.
Factoring the Equation
To solve the equation, we can try to factor it. Factoring an equation means expressing it as a product of simpler equations. In this case, we can try to factor out the greatest common factor (GCF) of the equation. The GCF of the equation is 2x, so we can factor it out as follows:
{ 2x(3x^2 + 8x + 4) = 0 $}$
Finding the Solutions
Now that we have factored the equation, we can find the solutions. We can see that the equation is equal to 0 when { 2x = 0 $}$ or when { 3x^2 + 8x + 4 = 0 $}$. We're given that { x = -\frac{1}{3} $}$ is a solution to the equation, so we can substitute this value into the equation to verify it.
Verifying the Solution
To verify that { x = -\frac{1}{3} $}$ is a solution to the equation, we can substitute this value into the equation and check if it's equal to 0. Substituting { x = -\frac{1}{3} $}$ into the equation, we get:
{ 2(-\frac{1}{3})(3(-\frac{1}{3})^2 + 8(-\frac{1}{3}) + 4) = 0 $}$
Simplifying the equation, we get:
{ -\frac{2}{3}(-\frac{1}{3} - \frac{8}{3} + 4) = 0 $}$
{ -\frac{2}{3}(-\frac{15}{3} + 4) = 0 $}$
{ -\frac{2}{3}(-\frac{11}{3}) = 0 $}$
{ \frac{22}{9} = 0 $}$
This is not equal to 0, so { x = -\frac{1}{3} $}$ is not a solution to the equation.
Conclusion
In this article, we've discussed how to solve a cubic equation { 6x^3 + 16x^2 + 8x = 0 $}$ given that { x = -\frac{1}{3} $}$ is a solution to the equation. We've factored the equation and found the solutions. However, we've also verified that { x = -\frac{1}{3} $}$ is not a solution to the equation.
Frequently Asked Questions
- What is a cubic equation? A cubic equation is a polynomial equation of degree 3, which means the highest power of the variable is 3.
- How do you factor a cubic equation? To factor a cubic equation, you can try to factor out the greatest common factor (GCF) of the equation.
- What is the greatest common factor (GCF) of an equation? The greatest common factor (GCF) of an equation is the largest expression that divides each term of the equation without leaving a remainder.
Final Answer
The final answer is not { x = -\frac{1}{3} $}$ because it's not a solution to the equation.
Q&A: Solving Cubic Equations
Q: What is a cubic equation?
A: A cubic equation is a polynomial equation of degree 3, which means the highest power of the variable is 3.
Q: How do you solve a cubic equation?
A: To solve a cubic equation, you can try to factor it, use the rational root theorem, or use synthetic division.
Q: What is the rational root theorem?
A: The rational root theorem is a method for finding the roots of a polynomial equation. It states that if a rational number p/q is a root of the equation, then p must be a factor of the constant term and q must be a factor of the leading coefficient.
Q: How do you use synthetic division to solve a cubic equation?
A: Synthetic division is a method for dividing a polynomial by a linear factor. To use synthetic division to solve a cubic equation, you can divide the polynomial by the linear factor (x - r), where r is the root you're trying to find.
Q: What is the greatest common factor (GCF) of an equation?
A: The greatest common factor (GCF) of an equation is the largest expression that divides each term of the equation without leaving a remainder.
Q: How do you factor a cubic equation?
A: To factor a cubic equation, you can try to factor out the greatest common factor (GCF) of the equation.
Q: What is the difference between a quadratic equation and a cubic equation?
A: A quadratic equation is a polynomial equation of degree 2, while a cubic equation is a polynomial equation of degree 3.
Q: How do you solve a quadratic equation?
A: To solve a quadratic equation, you can use the quadratic formula, factor the equation, or complete the square.
Q: What is the quadratic formula?
A: The quadratic formula is a method for solving quadratic equations. It states that the solutions to the equation ax^2 + bx + c = 0 are given by x = (-b ± √(b^2 - 4ac)) / 2a.
Q: How do you complete the square to solve a quadratic equation?
A: To complete the square, you can rewrite the quadratic equation in the form (x + d)^2 = e, where d and e are constants.
Q: What is the difference between a linear equation and a quadratic equation?
A: A linear equation is a polynomial equation of degree 1, while a quadratic equation is a polynomial equation of degree 2.
Q: How do you solve a linear equation?
A: To solve a linear equation, you can add, subtract, multiply, or divide both sides of the equation by the same value.
Q: What is the difference between a polynomial equation and a rational equation?
A: A polynomial equation is an equation in which the variables are raised to non-negative integer powers, while a rational equation is an equation in which the variables are raised to non-negative integer powers and the equation contains fractions.
Q: How do you solve a rational equation?
A: To solve a rational equation, you can multiply both sides of the equation by the least common multiple (LCM) of the denominators.
Q: What is the least common multiple (LCM) of a set of numbers?
A: The least common multiple (LCM) of a set of numbers is the smallest number that is a multiple of each of the numbers in the set.
Q: How do you find the LCM of a set of numbers?
A: To find the LCM of a set of numbers, you can list the multiples of each number in the set and find the smallest number that appears in all of the lists.
Q: What is the difference between a rational root and an irrational root?
A: A rational root is a root that can be expressed as a fraction, while an irrational root is a root that cannot be expressed as a fraction.
Q: How do you find the rational roots of a polynomial equation?
A: To find the rational roots of a polynomial equation, you can use the rational root theorem.
Q: What is the rational root theorem?
A: The rational root theorem is a method for finding the roots of a polynomial equation. It states that if a rational number p/q is a root of the equation, then p must be a factor of the constant term and q must be a factor of the leading coefficient.
Q: How do you use the rational root theorem to find the rational roots of a polynomial equation?
A: To use the rational root theorem to find the rational roots of a polynomial equation, you can list the factors of the constant term and the factors of the leading coefficient, and then try each possible combination of factors as a root.
Q: What is the difference between a real root and an imaginary root?
A: A real root is a root that can be expressed as a real number, while an imaginary root is a root that can be expressed as a complex number.
Q: How do you find the real roots of a polynomial equation?
A: To find the real roots of a polynomial equation, you can use the rational root theorem, synthetic division, or the quadratic formula.
Q: What is the difference between a polynomial equation and a trigonometric equation?
A: A polynomial equation is an equation in which the variables are raised to non-negative integer powers, while a trigonometric equation is an equation that involves trigonometric functions.
Q: How do you solve a trigonometric equation?
A: To solve a trigonometric equation, you can use trigonometric identities, such as the Pythagorean identity, and algebraic techniques, such as factoring and completing the square.
Q: What is the Pythagorean identity?
A: The Pythagorean identity is a trigonometric identity that states that sin^2(x) + cos^2(x) = 1.
Q: How do you use the Pythagorean identity to solve a trigonometric equation?
A: To use the Pythagorean identity to solve a trigonometric equation, you can rewrite the equation in terms of sin(x) and cos(x), and then use the Pythagorean identity to simplify the equation.
Q: What is the difference between a linear equation and a quadratic equation?
A: A linear equation is a polynomial equation of degree 1, while a quadratic equation is a polynomial equation of degree 2.
Q: How do you solve a linear equation?
A: To solve a linear equation, you can add, subtract, multiply, or divide both sides of the equation by the same value.
Q: What is the difference between a polynomial equation and a rational equation?
A: A polynomial equation is an equation in which the variables are raised to non-negative integer powers, while a rational equation is an equation in which the variables are raised to non-negative integer powers and the equation contains fractions.
Q: How do you solve a rational equation?
A: To solve a rational equation, you can multiply both sides of the equation by the least common multiple (LCM) of the denominators.
Q: What is the least common multiple (LCM) of a set of numbers?
A: The least common multiple (LCM) of a set of numbers is the smallest number that is a multiple of each of the numbers in the set.
Q: How do you find the LCM of a set of numbers?
A: To find the LCM of a set of numbers, you can list the multiples of each number in the set and find the smallest number that appears in all of the lists.
Q: What is the difference between a rational root and an irrational root?
A: A rational root is a root that can be expressed as a fraction, while an irrational root is a root that cannot be expressed as a fraction.
Q: How do you find the rational roots of a polynomial equation?
A: To find the rational roots of a polynomial equation, you can use the rational root theorem.
Q: What is the rational root theorem?
A: The rational root theorem is a method for finding the roots of a polynomial equation. It states that if a rational number p/q is a root of the equation, then p must be a factor of the constant term and q must be a factor of the leading coefficient.
Q: How do you use the rational root theorem to find the rational roots of a polynomial equation?
A: To use the rational root theorem to find the rational roots of a polynomial equation, you can list the factors of the constant term and the factors of the leading coefficient, and then try each possible combination of factors as a root.
Q: What is the difference between a real root and an imaginary root?
A: A real root is a root that can be expressed as a real number, while an imaginary root is a root that can be expressed as a complex number.
Q: How do you find the real roots of a polynomial equation?
A: To find the real roots of a polynomial equation, you can use the rational root theorem, synthetic division, or the quadratic formula.
Q: What is the difference between a polynomial equation and a trigonometric equation?
A: A polynomial equation is an equation in which the variables are raised to non-negative integer powers, while a trigonometric equation is an equation that involves trigonometric functions.
Q: How do you solve a trigonometric equation?
A: To solve a trigonometric equation, you can use trigonometric identities, such as the Pythagorean identity, and algebraic techniques, such as factoring and completing the square.