Factor The Following Polynomial:$\[ 36x^3 - 132x^2 + 121x \\]Given: $\[ x(6x - [?])^2 \\]

Introduction

Factoring polynomials is an essential skill in algebra, and it can be a challenging task, especially when dealing with complex expressions. In this article, we will focus on factoring the given polynomial $36x^3 - 132x^2 + 121x$. We will use the given expression $x(6x - [?])^2$ to guide us in factoring the polynomial.

Understanding the Given Expression

The given expression is $x(6x - [?])^2$. To factor the polynomial, we need to identify the value of $[?]$ that makes the expression true. Let's start by expanding the expression:

$$x(6x - [?])^2 = x(36x^2 - 2 \cdot 6x \cdot [?] + [?]^2)$$

Identifying the Value of [?]

To factor the polynomial, we need to identify the value of $[?]$ that makes the expression true. Let's analyze the given polynomial $36x^3 - 132x^2 + 121x$. We can see that the polynomial has a common factor of $x$, so we can rewrite it as:

$$36x^3 - 132x^2 + 121x = x(36x^2 - 132x + 121)$$

Factoring the Quadratic Expression

Now that we have identified the value of $[?]$ as $121$, we can factor the quadratic expression $36x^2 - 132x + 121$. We can start by finding the factors of $121$ that add up to $-132$. The factors of $121$ are $1, 11, 121$. We can see that $-11$ and $-11$ add up to $-22$, which is close to $-132$. However, we need to find the correct factors.

Using the Quadratic Formula

To factor the quadratic expression, we can use the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 36$, $b = -132$, and $c = 121$. Plugging these values into the formula, we get:

$$x = \frac{-(-132) \pm \sqrt{(-132)^2 - 4(36)(121)}}{2(36)}$$

Simplifying the Expression

Simplifying the expression, we get:

$$x = \frac{132 \pm \sqrt{17424 - 21104}}{72}$$

$$x = \frac{132 \pm \sqrt{-3660}}{72}$$

Identifying the Complex Roots

The expression under the square root is negative, which means that the quadratic expression has complex roots. The complex roots are:

$$x = \frac{132 \pm i\sqrt{3660}}{72}$$

Factoring the Quadratic Expression

Now that we have identified the complex roots, we can factor the quadratic expression $36x^2 - 132x + 121$. We can write it as:

$$36x^2 - 132x + 121 = (6x - 11)^2$$

Factoring the Polynomial

Now that we have factored the quadratic expression, we can factor the polynomial $36x^3 - 132x^2 + 121x$. We can write it as:

$$36x^3 - 132x^2 + 121x = x(6x - 11)^2$$

Conclusion

In this article, we have factored the given polynomial $36x^3 - 132x^2 + 121x$. We used the given expression $x(6x - [?])^2$ to guide us in factoring the polynomial. We identified the value of $[?]$ as $121$ and factored the quadratic expression $36x^2 - 132x + 121$. We used the quadratic formula to find the complex roots and factored the quadratic expression as $(6x - 11)^2$. Finally, we factored the polynomial as $x(6x - 11)^2$.

Final Answer

The final answer is $x(6x - 11)^2$.

$$

Introduction

In our previous article, we factored the polynomial $36x^3 - 132x^2 + 121x$ using the given expression $x(6x - [?])^2$. We identified the value of $[?]$ as $121$ and factored the quadratic expression $36x^2 - 132x + 121$. In this article, we will answer some common questions related to factoring the polynomial.

Q: What is the first step in factoring the polynomial?

A: The first step in factoring the polynomial is to identify the value of $[?]$ in the given expression $x(6x - [?])^2$. This will help us to factor the quadratic expression.

Q: How do I identify the value of $[?]$?

A: To identify the value of $[?]$, we need to analyze the given polynomial $36x^3 - 132x^2 + 121x$. We can see that the polynomial has a common factor of $x$, so we can rewrite it as $x(36x^2 - 132x + 121)$. We can then identify the value of $[?]$ as $121$.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to find the roots of a quadratic equation. It is given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: How do I use the quadratic formula to factor the quadratic expression?

A: To use the quadratic formula to factor the quadratic expression, we need to plug in the values of $a$, $b$, and $c$ into the formula. In this case, $a = 36$, $b = -132$, and $c = 121$. We can then simplify the expression to find the roots of the quadratic equation.

Q: What are the roots of the quadratic equation?

A: The roots of the quadratic equation are the values of $x$ that make the equation true. In this case, the roots are complex numbers, given by:

$$x = \frac{132 \pm i\sqrt{3660}}{72}$$

Q: How do I factor the quadratic expression using the roots?

A: To factor the quadratic expression using the roots, we can write it as:

$$36x^2 - 132x + 121 = (6x - 11)^2$$

Q: What is the final answer to the factored polynomial?

A: The final answer to the factored polynomial is $x(6x - 11)^2$.

Q: Can I use other methods to factor the polynomial?

A: Yes, there are other methods that can be used to factor the polynomial, such as grouping and synthetic division. However, the method used in this article is a common and effective way to factor the polynomial.

Q: What are some common mistakes to avoid when factoring polynomials?

A: Some common mistakes to avoid when factoring polynomials include:

• Not identifying the value of $[?]$ correctly
• Not using the quadratic formula correctly
• Not simplifying the expression correctly
• Not factoring the quadratic expression correctly

Conclusion

In this article, we have answered some common questions related to factoring the polynomial $36x^3 - 132x^2 + 121x$. We have provided step-by-step instructions on how to factor the polynomial using the given expression $x(6x - [?])^2$. We have also discussed some common mistakes to avoid when factoring polynomials.

Final Answer

The final answer to the factored polynomial is $x(6x - 11)^2$.