One root of f(x) = 2×3 + 9×2 + 7x – 6 is –3. explain how to find the factors of the polynomial
Understanding and solving polynomials is a fundamental aspect of algebra, and one intriguing polynomial that often appears in mathematical studies is f(x) = 2x^3 + 9x^2 + 7x – 6. In this exploration, we will delve into the process of solving for the root when x equals -3, unraveling the intricate web of factors that constitute the polynomial.
The Foundation: Polynomial Basics
Before we embark on solving the equation, let’s establish a solid foundation by revisiting some polynomial basics. A polynomial is a mathematical expression consisting of variables, coefficients, and exponents, and it often plays a pivotal role in various mathematical models and real-world scenarios. The power of the polynomial, represented by the highest exponent, influences its behavior and the number of roots it possesses.
Breaking Down the Polynomial: f(x) = 2x^3 + 9x^2 + 7x – 6
Our focus lies in the polynomial f(x) = 2x^3 + 9x^2 + 7x – 6, and our goal is to find the root when x is set to -3. To begin, we need to understand the anatomy of the polynomial. The highest exponent here is 3, indicating that this is a cubic polynomial. Cubic polynomials can have up to three roots, and our task is to pinpoint one of them when x is -3.
Expressing the Equation: f(-3) = 2(-3)^3 + 9(-3)^2 + 7(-3) – 6
To find the root at x = -3, substitute -3 into the equation for x:
�(−3)=2(−3)3+9(−3)2+7(−3)–6
f(−3)=2(−3)
3
+9(−3)
2
+7(−3)–6
This expression may seem complex, but fear not; we will break it down step by step.
Step 1: Compute the Powers
Firstly, calculate the powers of -3:
(−3)3=−27
(−3)
3
=−27
(−3)2=9
(−3)
2
=9
Step 2: Substitute and Simplify
Now, substitute these values back into the original equation:
�(−3)=2(−27)+9(9)+7(−3)–6
f(−3)=2(−27)+9(9)+7(−3)–6
Step 3: Simplify Further
Continue simplifying:
�(−3)=−54+81−21–6
f(−3)=−54+81−21–6
�(−3)=0
f(−3)=0
Surprisingly, the result is zero! This implies that -3 is indeed a root of the polynomial. But the journey doesn’t end here; we must delve deeper to understand the factors that lead to this outcome.
Factorizing the Polynomial
To comprehend how -3 became a root, we must factorize the polynomial. Factorization involves breaking down a polynomial into its constituent factors, revealing the roots more explicitly.
Synthetic Division: Unveiling the Factors
One method to factorize polynomials efficiently is synthetic division. Using synthetic division with -3 as the root, we can uncover the quotient and remainders that expose the factors.
2�3+9�2+7�–6 divided by (�+3)
2x
3
+9x
2
+7x–6divided by(x+3)
Let’s perform synthetic division:
-3 | 2 9 7 -6
|__________
| 0 -6 -48
The result indicates that the quotient is
2�2−6�−48
2x
2
−6x−48. Now, this quadratic expression represents the factorized form of the original polynomial.
Solving the Quadratic Equation
To find the remaining factors, we need to solve the quadratic equation
2�2−6�−48=0
2x
2
−6x−48=0. This equation can be factored into
(�−6)(2�+8)=0
(x−6)(2x+8)=0, yielding the factors
(�−6)
(x−6) and
(2�+8)
(2x+8).
The Final Insight: Understanding the Roots
Combining all factors, the original polynomial
2�3+9�2+7�–6
2x
3
+9x
2
+7x–6 can be expressed as
(�+3)(�−6)(2�+8)
(x+3)(x−6)(2x+8). Now, it becomes evident why -3 is a root; when
(�+3)
(x+3) is set to zero,
�
x equals -3, confirming our initial observation.
Conclusion
In this journey of unraveling polynomial roots, we’ve navigated through the intricacies of
�(�)=2�3+9�2+7�–6
f(x)=2x
3
+9x
2
+7x–6, specifically when x is -3. From the initial substitution to the synthetic division and factoring, each step has contributed to our understanding of how -3 became a root and how the polynomial can be expressed in its factored form. This exploration not only enhances our grasp of algebraic concepts but also provides a roadmap for tackling similar polynomial challenges in the future.