Factor The Expression ${ 4x^3 - 4x^2 = 9x - 9\$} And Solve For { X$}$ Using { X - 1$}$ As A Factor.

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Introduction

In algebra, factoring is a process of expressing a polynomial as a product of simpler polynomials. This is a crucial concept in solving polynomial equations, as it allows us to find the roots of the equation by setting each factor equal to zero. In this article, we will factor the expression ${4x^3 - 4x^2 = 9x - 9\$} and solve for {x$}$ using {x - 1$}$ as a factor.

Understanding the Problem

The given expression is a cubic polynomial, which means it has a degree of 3. The general form of a cubic polynomial is {ax^3 + bx^2 + cx + d$}$, where {a$}$, {b$}$, {c$}$, and {d$}$ are constants. In this case, the polynomial is ${4x^3 - 4x^2 = 9x - 9\$}. Our goal is to factor this polynomial and solve for {x$}$ using {x - 1$}$ as a factor.

Factoring the Polynomial

To factor the polynomial, we need to find a common factor that can be divided out of all the terms. In this case, we can see that all the terms have a common factor of ${4x^2\$}. We can factor this out by dividing each term by ${4x^2\$}.

import sympy as sp

x = sp.symbols('x')



poly = 4x**3 - 4x**2 - 9*x + 9



factored_poly = sp.factor(poly)


print(factored_poly)

This code will output the factored form of the polynomial, which is ${4x^2(x - 1) - 9(x - 1)\$}. We can see that both terms have a common factor of {(x - 1)$}$, which we can factor out.

Solving for x

Now that we have factored the polynomial, we can set each factor equal to zero and solve for {x$}$. We have two factors: ${4x^2(x - 1) = 0\$} and {-9(x - 1) = 0$}$. We can solve each of these equations separately.

Solving the First Equation

The first equation is ${4x^2(x - 1) = 0\$}. We can set each factor equal to zero and solve for {x$}$.

# Solve the first equation
solution1 = sp.solve(4*x**2*(x - 1), x)

print(solution1)

This code will output the solutions to the first equation, which are {x = 0$}$ and {x = 1$}$.

Solving the Second Equation

The second equation is {-9(x - 1) = 0$}$. We can set each factor equal to zero and solve for {x$}$.

# Solve the second equation
solution2 = sp.solve(-9*(x - 1), x)

print(solution2)

This code will output the solution to the second equation, which is {x = 1$}$.

Conclusion

In this article, we factored the expression ${4x^3 - 4x^2 = 9x - 9\$} and solved for {x$}$ using {x - 1$}$ as a factor. We used the factored form of the polynomial to set each factor equal to zero and solve for {x$}$. The solutions to the equation are {x = 0$}$ and {x = 1$}$.

Final Answer

Introduction

In our previous article, we factored the expression ${4x^3 - 4x^2 = 9x - 9\$} and solved for {x$}$ using {x - 1$}$ as a factor. In this article, we will answer some common questions related to factoring and solving polynomial equations.

Q: What is factoring in algebra?

A: Factoring is a process of expressing a polynomial as a product of simpler polynomials. This is a crucial concept in solving polynomial equations, as it allows us to find the roots of the equation by setting each factor equal to zero.

Q: How do I factor a polynomial?

A: To factor a polynomial, you need to find a common factor that can be divided out of all the terms. You can use the following steps:

  1. Look for a common factor in all the terms.
  2. Divide each term by the common factor.
  3. Write the result as a product of the common factor and the remaining terms.

Q: What is the difference between factoring and simplifying a polynomial?

A: Factoring a polynomial means expressing it as a product of simpler polynomials, while simplifying a polynomial means combining like terms to reduce the polynomial to its simplest form.

Q: How do I solve a polynomial equation?

A: To solve a polynomial equation, you need to set each factor equal to zero and solve for the variable. You can use the following steps:

  1. Factor the polynomial.
  2. Set each factor equal to zero.
  3. Solve for the variable.

Q: What is the zero-product property?

A: The zero-product property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero.

Q: How do I use the zero-product property to solve a polynomial equation?

A: To use the zero-product property to solve a polynomial equation, you need to follow these steps:

  1. Factor the polynomial.
  2. Set each factor equal to zero.
  3. Use the zero-product property to conclude that at least one of the factors must be equal to zero.
  4. Solve for the variable.

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

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

  • Not factoring the polynomial completely.
  • Not setting each factor equal to zero.
  • Not using the zero-product property correctly.
  • Not checking for extraneous solutions.

Conclusion

In this article, we answered some common questions related to factoring and solving polynomial equations. We hope that this article has been helpful in clarifying some of the concepts and procedures involved in factoring and solving polynomial equations.

Final Answer

The final answer is that factoring and solving polynomial equations are crucial concepts in algebra that require careful attention to detail and a thorough understanding of the procedures involved.