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

Polynomial factorization is a crucial skill in algebra, allowing mathematicians to break down complex equations into simpler components and unveil the roots of the polynomial. In this exploration, we’ll navigate the intricate process of polynomial factorization, focusing on the polynomial equation

�(�)=2�3+9�2+7�–6

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

3

+9*x*

2

+7*x*–6 and decoding its secrets at

�=−3

*x*=−3.

**Understanding Polynomial Factorization**

Polynomial factorization involves expressing a polynomial as the product of its factors. These factors, when multiplied together, yield the original polynomial equation. Factorization plays a pivotal role in solving equations, analyzing functions, and uncovering insights into mathematical models.

**Exploring the Polynomial Equation**

Before we embark on the journey of factorization, 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 cubic polynomial presents a challenge in deciphering its roots, but through systematic methods, we can navigate its complexities and unlock its secrets.

**Deciphering the Polynomial at**

### �=−3

*x*=−3

Our goal is to understand the behavior 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 evaluating the polynomial at

�=−3

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

#### Step 1: Substituting

#### �=−3

*x*=−3 into the Polynomial

Let’s substitute

�=−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 suggests 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, indicating the presence of a factor

(�+3)

(*x*+3).

**Utilizing Synthetic Division**

Synthetic division is a powerful tool for factorizing polynomials and identifying roots. Let’s employ synthetic division to divide the polynomial by

(�+3)

(*x*+3), confirming its status as a factor and revealing additional insights into the polynomial’s structure.

-3 | 2 9 7 -6

|__________

| 0 0 0 0

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 roots and factors.

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

With

(�+3)

(*x*+3) identified 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)

**Further Analysis: Factoring the Quadratic Expression**

Now that we’ve factorized the polynomial, let’s delve deeper into the quadratic expression

2�2+3�–2

2*x*

2

+3*x*–2 to uncover its roots and gain a comprehensive understanding of the polynomial’s behavior.

**Conclusion: Navigating the Complexity**

In this exploration, we’ve navigated the complexities of polynomial factorization, cracking the code 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. Through systematic methods such as synthetic division and factorization, we’ve unraveled the polynomial’s roots and factors, shedding light on its structure and behavior. Mastery of polynomial factorization empowers mathematicians to solve equations, analyze functions, and unlock the mysteries of mathematics.