Understanding Polynomials
Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. They are often used to represent real-world problems and have a wide range of applications in mathematics, physics, and engineering.
Factoring Polynomials
Factoring a polynomial means expressing it as the product of other polynomials. The factored form of a polynomial allows us to find its roots, solve equations, and simplify calculations. In many cases, it is important to factor polynomials completely to fully understand their behavior and make further calculations easier.
The Given Polynomial: 4 4X 4 1
The given polynomial, 4x⁴ + 4x² + 1, is a fourth-degree polynomial with terms of varying powers of x. In order to determine if it is factored completely, we need to follow a systematic approach to factor it. Let’s break it down step by step.
Factoring Process
Factoring a polynomial involves several methods, including factoring by grouping, factoring trinomials, factoring special cases, and factoring by using the sum or difference of cubes. In this case, we will utilize the method of factoring a sum of two perfect squares to factor the given polynomial completely.
Factoring Using the Sum of Two Squares
The given polynomial, 4x⁴ + 4x² + 1, can be factored using the sum of two perfect squares formula, which states that a² + 2ab + b² = (a+b)². In this case, we need to express the polynomial as a sum of two perfect squares. We notice that 4x⁴ and 1 are perfect squares, and 4x² is twice their product. Therefore, we need to express 4x⁴ + 4x² + 1 in a form that resembles the sum of two perfect squares.
Utilizing Substitution
To simplify the factoring process, we can use a substitution to express the given polynomial in terms of a new variable. Let’s set y = x². This gives us 4y² + 4y + 1, which looks more like a quadratic trinomial that can be factored easily.
Factoring the Substituted Polynomial
With the substitution y = x², the polynomial becomes 4y² + 4y + 1, which can be factored using the standard techniques for factoring quadratic trinomials. When factoring, we look for two numbers that multiply to the constant term and add to the coefficient of the middle term. In this case, those numbers are 1 and 1, and the factored form of 4y² + 4y + 1 is (2y + 1)².
Back Substitution
Now that we have factored the polynomial in terms of y, we can substitute back x² for y to restore the original variable. This gives us the factored form of the given polynomial as (2x² + 1)².
Verifying the Factored Form
To confirm that the polynomial has been factored completely, we can expand the factored form to check if it yields the original polynomial. (2x² + 1)² can be expanded using the formula for the square of a binomial: (a + b)² = a² + 2ab + b². Applying this formula yields 4x⁴ + 4x² + 1, which matches the original polynomial. Therefore, we have successfully factored the given polynomial completely as (2x² + 1)².
Conclusion
Factoring polynomials is a fundamental aspect of algebra and provides valuable insights into the behavior and properties of mathematical expressions. In the case of the polynomial 4x⁴ + 4x² + 1, we employed the sum of two perfect squares method to factor it completely as (2x² + 1)². This factored form allows for easier analysis, solving equations, and understanding the roots of the polynomial.
