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

Polynomials are fundamental mathematical expressions that appear in various contexts, from algebra to calculus. Understanding how to decode polynomial roots is crucial for solving equations, analyzing functions, and uncovering valuable insights into mathematical models. In this comprehensive guide, we will unravel the step-by-step method to factorize the polynomial equation

�(�)=2�3+9�2+7�–6

f(x)=2x

3

+9x

2

+7x–6, shedding light on the secrets of polynomial roots and factorization.

Unveiling Polynomial Roots

Before we dive into the factorization process, let’s elucidate the concept of polynomial roots. The roots of a polynomial are the values of

x that make the polynomial equal to zero. These roots are crucial in understanding the behavior of the polynomial function, including its intercepts, turning points, and end behavior. Decoding polynomial roots involves identifying these critical values and unraveling the factors that contribute to their existence.

Exploring the Polynomial Equation

Our journey begins with the polynomial equation

�(�)=2�3+9�2+7�–6

f(x)=2x

3

+9x

2

+7x–6. This polynomial, known as a cubic polynomial due to its highest power of

x being three, presents an intriguing challenge in the quest to decode its roots. By dissecting this polynomial and applying systematic methods, we can unlock its secrets and reveal the factors that contribute to its structure.

Step-by-Step Method for Factorization

Factorizing a polynomial involves breaking it down into simpler expressions, known as its factors, which when multiplied together yield the original polynomial. Let’s outline a step-by-step method for factorizing the polynomial

�(�)=2�3+9�2+7�–6

f(x)=2x

3

+9x

2

+7x–6 to decode its roots:

Step 1: Identify Common Factors

Begin by identifying any common factors among the terms of the polynomial. Common factors can be constants or variables that divide each term evenly. Factoring out common factors simplifies the polynomial expression, making the factorization process more manageable.

Step 2: Explore Factorization Techniques

Once common factors are identified, explore various factorization techniques to break down the polynomial further. Techniques such as factoring by grouping, factoring trinomials, and using synthetic division are commonly employed to factorize polynomials. Choose the most suitable technique based on the structure of the polynomial.

Step 3: Apply Factorization Methods

Apply the chosen factorization method to the polynomial equation

�(�)=2�3+9�2+7�–6

f(x)=2x

3

+9x

2

+7x–6. This step may involve trial and error, as well as algebraic manipulation to simplify expressions and identify factors. The goal is to express the polynomial as a product of its factors.

Step 4: Check for Rational Roots

Utilize the rational root theorem or synthetic division to identify potential rational roots of the polynomial equation. Rational roots are crucial in factorizing polynomials and can provide valuable insights into their structure. Verify potential roots by substituting them into the polynomial equation and checking if they yield zero.

Step 5: Verify Factorization

Once potential factors are identified, verify the factorization by multiplying the factors together to ensure they yield the original polynomial expression. This validation step confirms the accuracy of the factorization process and ensures that no factors have been overlooked.

Factorization of

�(�)=2�3+9�2+7�–6

f(x)=2x

3

+9x

2

+7x–6

Now, let’s apply the step-by-step method outlined above to factorize the polynomial equation

�(�)=2�3+9�2+7�–6

f(x)=2x

3

+9x

2

+7x–6.

Step 1: Identify Common Factors

Upon inspection, there are no common factors among the terms of the polynomial.

Step 2: Explore Factorization Techniques

Given that the polynomial is cubic, we’ll explore factoring by grouping or using synthetic division to identify potential roots.

Step 3: Apply Factorization Methods

Let’s attempt factoring by grouping:

2�3+9�2+7�–6

2x

3

+9x

2

+7x–6

=(2�3+9�2)+(7�–6)

=(2x

3

+9x

2

)+(7x–6)

=�2(2�+9)+1(7�–6)

=x

2

(2x+9)+1(7x–6)

Step 4: Check for Rational Roots

Applying the rational root theorem, we identify potential rational roots as factors of the constant term divided by factors of the leading coefficient:

±1,±2,±3,±6

±1,±2,±3,±6

Step 5: Verify Factorization

Synthetic division or polynomial multiplication can be used to verify the factorization obtained.

Conclusion: Deciphering Polynomial Roots

In this comprehensive guide, we’ve decoded the polynomial roots by unraveling the step-by-step method to factorize the polynomial equation

�(�)=2�3+9�2+7�–6

f(x)=2x

3

+9x

2

+7x–6. By applying systematic techniques and exploring various factorization methods, we’ve unveiled the factors that contribute to the polynomial’s structure and identified potential roots. Mastery of polynomial roots empowers mathematicians and scientists to solve equations, analyze functions, and unlock the mysteries of mathematics.