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

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

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

Polynomials are mathematical expressions consisting of variables and coefficients, often encountered in algebraic equations and real-world problems. Understanding polynomial factors and their significance is essential in various mathematical applications, from solving equations to analyzing functions. In this exploration, we will delve into the secrets of polynomial factors, focusing on the polynomial equation

�(�)=2�3+9�2+7�–6

f(x)=2x

3

+9x

2

+7x–6 and its behavior at

�=−3

x=−3.

Unraveling the Polynomial Equation

Before we delve into the intricacies of polynomial factors, let’s familiarize ourselves with the polynomial equation

�(�)=2�3+9�2+7�–6

f(x)=2x

3

+9x

2

+7x–6. This polynomial is of degree three, known as a cubic polynomial, owing to the highest power of the variable

x being three. Each term in the polynomial consists of a coefficient multiplied by a power of

x, with the constant term being -6.

Significance of Polynomial Factors

Polynomial factors are the expressions that divide the polynomial evenly, revealing its roots and aiding in solving equations. Understanding polynomial factors provides insights into the behavior of the polynomial function, including its intercepts, end behavior, and turning points. By unlocking the secrets of polynomial factors, mathematicians and scientists gain valuable tools for solving problems and making predictions.

Analyzing

�(�)=2�3+9�2+7�–6

f(x)=2x

3

+9x

2

+7x–6 at

�=−3

x=−3

Now, let’s focus on understanding the polynomial equation

�(�)=2�3+9�2+7�–6

f(x)=2x

3

+9x

2

+7x–6 at

�=−3

x=−3. This involves evaluating the polynomial at

�=−3

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

Step 1: Substituting

�=−3

x=−3 into the Polynomial

To analyze the polynomial at

�=−3

x=−3, we substitute

�=−3

x=−3 into the polynomial equation:

�(−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

Let’s compute the value of

�(−3)

f(−3) to understand the polynomial’s behavior at

�=−3

x=−3:

�(−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)=2x

3

+9x

2

+7x–6. In other words, when

�=−3

x=−3, the polynomial evaluates to zero, suggesting that

(�+3)

(x+3) is a factor of the polynomial.

Factorizing the Polynomial

Understanding that

�=−3

x=−3 is a root of the polynomial equation allows us to factorize the polynomial and express it as the product of its factors. Factorization reveals the roots of the polynomial and simplifies solving equations involving the polynomial.

Factoring Using Synthetic Division

One method for factorizing polynomials is synthetic division. Let’s utilize synthetic division to divide the polynomial by

(�+3)

(x+3), confirming that

(�+3)

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

-3 | 2    9    7   -6

|__________

|  0    0    0    0

 

The result of synthetic division confirms that

(�+3)

(x+3) evenly divides the polynomial, with a remainder of zero. Therefore,

(�+3)

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

�(�)=2�3+9�2+7�–6

f(x)=2x

3

+9x

2

+7x–6.

Expressing the Polynomial as a Product of Factors

Since

(�+3)

(x+3) is a factor of the polynomial, we can express the polynomial as a product of its factors:

�(�)=(�+3)(2�2+3�–2)

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

2

+3x–2)

Solving the Quadratic Equation

Now that we have factored the polynomial, we can further explore the quadratic factor

2�2+3�–2

2x

2

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

 

Conclusion: Unveiling Polynomial Secrets

In this exploration, we’ve unlocked the secrets of polynomial factors by understanding the polynomial equation

�(�)=2�3+9�2+7�–6

f(x)=2x

3

+9x

2

+7x–6 at

�=−3

x=−3. By analyzing the polynomial at

�=−3

x=−3, we determined that

(�+3)

(x+3) is a factor of the polynomial, leading to its factorization and revealing additional insights into its roots. Through this process, we’ve gained a deeper understanding of polynomial behavior and the significance of polynomial factors in solving equations and analyzing functions.