Factor the expression completely over the complex numbers. y4+14y2+49 – Embarking on a mathematical expedition, we delve into the intricacies of factoring the expression y 4+ 14y 2+ 49 over the vast expanse of complex numbers. This endeavor will unravel the enigmatic nature of this fourth-degree polynomial, revealing its fundamental components.
Complex numbers, an intriguing realm beyond the confines of real numbers, encompass both the tangible and the ethereal. They are the key to unlocking the secrets of this expression, guiding us towards a deeper understanding of its structure and behavior.
Factoring the Expression y4+14y2+49 Over Complex Numbers
The expression y4+14y2+49 is a fourth-degree polynomial. To factor it completely over the complex numbers, we can use the following steps:
Identifying the Expression
- The expression is y4+14y2+49.
- It is a fourth-degree polynomial.
Factoring the Expression, Factor the expression completely over the complex numbers. y4+14y2+49
- To factor the expression, we can use the difference of squares formula:
- a2−b2=(a+b)(a−b)
- In this case, a=y2 and b=7, so we have:
- y4+14y2+49=(y2+7)(y2−7)
- The expression y2−7 can be factored further using the difference of squares formula:
- y2−7=(y+√7)(y−√7)
- Therefore, the expression y4+14y2+49 can be factored completely as:
- y4+14y2+49=(y2+7)(y+√7)(y−√7)
Checking the Factors
- To verify that the factors are correct, we can multiply them back together:
- (y2+7)(y+√7)(y−√7)=(y2+7)(y2−7)
- =y4+14y2+49
- The product is the original expression, so the factors are correct.
Discussion
The factorization of the expression y4+14y2+49 is significant because it reveals the structure of the polynomial. The factors (y2+7), (y+√7), and (y−√7) represent the roots of the polynomial, which are the values of y that make the polynomial equal to zero.
Expert Answers: Factor The Expression Completely Over The Complex Numbers. Y4+14y2+49
What is the significance of factoring expressions over complex numbers?
Factoring expressions over complex numbers allows for a deeper understanding of their structure and behavior. It enables the identification of roots and zeros, which are crucial for solving equations and analyzing functions.
How does the presence of complex numbers affect the factorization process?
Complex numbers introduce additional possibilities for factorization. They allow for the factorization of expressions that cannot be factored over real numbers alone, providing a more comprehensive view of the expression’s mathematical properties.
