## One root of f(x) = 2×3 + 9×2 + 7x – 6 is –3. explain how to find the factors of the polynomial

Polynomials serve as the backbone of algebra, appearing in numerous mathematical problems and real-world scenarios. Understanding their behavior and uncovering their roots is essential for solving equations and gaining insights into mathematical models. In this exploration, we embark on a quest to investigate the factors of the polynomial equation

�(�)=2�3+9�2+7�–6

*f*(*x*)=2*x*

3

+9*x*

2

+7*x*–6 at

�=−3

*x*=−3, delving into the intricacies of polynomial solutions and their significance.

**Exploring Polynomial Solutions**

Polynomial solutions, also known as roots, are the values of

�

*x* that make the polynomial equation equal to zero. These solutions play a crucial role in algebraic equations, providing key insights into the behavior and properties of polynomial functions. Investigating polynomial solutions involves identifying these critical values and understanding their implications.

**Understanding the Polynomial Equation**

Before we delve into the investigation of polynomial solutions, let’s acquaint ourselves with the polynomial equation

�(�)=2�3+9�2+7�–6

*f*(*x*)=2*x*

3

+9*x*

2

+7*x*–6. This polynomial, characterized by its degree and coefficients, presents an intriguing challenge in our quest for solutions. By dissecting its components and employing systematic methods, we can unravel its secrets and uncover its factors.

**Investigating the Factors at**

### �=−3

*x*=−3

Our quest focuses on investigating the factors of the polynomial equation

�(�)=2�3+9�2+7�–6

*f*(*x*)=2*x*

3

+9*x*

2

+7*x*–6 at

�=−3

*x*=−3. This involves analyzing the polynomial expression at

�=−3

*x*=−3 to determine its value and gain insights into its behavior at that specific point.

#### Step 1: Substituting

#### �=−3

*x*=−3 into the Polynomial

Let’s begin our investigation by substituting

�=−3

*x*=−3 into the polynomial equation

�(�)=2�3+9�2+7�–6

*f*(*x*)=2*x*

3

+9*x*

2

+7*x*–6:

�(−3)=2(−3)3+9(−3)2+7(−3)–6

*f*(−3)=2(−3)

3

+9(−3)

2

+7(−3)–6

#### Step 2: Computing the Value

Computing the value of

�(−3)

*f*(−3) yields:

�(−3)=2(−27)+9(9)−21–6

*f*(−3)=2(−27)+9(9)−21–6

�(−3)=−54+81−21–6

*f*(−3)=−54+81−21–6

�(−3)=0

*f*(−3)=0

The result

�(−3)=0

*f*(−3)=0 indicates that

�=−3

*x*=−3 is a root of the polynomial equation

�(�)=2�3+9�2+7�–6

*f*(*x*)=2*x*

3

+9*x*

2

+7*x*–6. In other words, when

�=−3

*x*=−3, the polynomial evaluates to zero, suggesting the presence of a factor

(�+3)

(*x*+3).

**Unraveling the Polynomial Factors**

With

�=−3

*x*=−3 identified as a root, our quest shifts to unraveling the factors of the polynomial equation

�(�)=2�3+9�2+7�–6

*f*(*x*)=2*x*

3

+9*x*

2

+7*x*–6. By understanding its factors, we can gain deeper insights into its structure and behavior.

#### Synthetic Division: Confirming Factors

Synthetic division is a powerful tool for confirming factors and unraveling polynomial solutions. Let’s utilize synthetic division to divide the polynomial by

(�+3)

(*x*+3) and confirm its status as a factor.

The result of synthetic division confirms that

(�+3)

(*x*+3) is indeed a factor of the polynomial equation

�(�)=2�3+9�2+7�–6

*f*(*x*)=2*x*

3

+9*x*

2

+7*x*–6, with a remainder of zero. This validation strengthens our understanding of the polynomial’s factors and roots.

**Expressing the Polynomial as a Product of Factors**

With

(�+3)

(*x*+3) confirmed as a factor, we can express the polynomial equation

�(�)=2�3+9�2+7�–6

*f*(*x*)=2*x*

3

+9*x*

2

+7*x*–6 as a product of its factors:

�(�)=(�+3)(2�2+3�–2)

*f*(*x*)=(*x*+3)(2*x*

2

+3*x*–2)

**Investigating the Quadratic Expression**

Further investigation is warranted to analyze the quadratic expression

2�2+3�–2

2*x*

2

+3*x*–2 and uncover its roots. By solving the quadratic equation, we can identify additional factors and gain a comprehensive understanding of the polynomial’s behavior.

**Conclusion: The Journey Continues**

In our quest for polynomial solutions, we’ve investigated the factors of the polynomial equation

�(�)=2�3+9�2+7�–6

*f*(*x*)=2*x*

3

+9*x*

2

+7*x*–6 at

�=−3

*x*=−3, unraveling its secrets and uncovering its structure. By employing systematic methods and analytical techniques, we’ve gained valuable insights into polynomial solutions and their significance in algebraic equations. As the journey continues, we deepen our understanding of polynomial functions and their role in mathematical exploration.