Problem 26 - Calculus Single Variable

Question

Write the given polynomial as a product of irreducible polynomials of degree one or two. \(x^{4}+3 x^{2}+2\)

Step-by-Step Solution

Answer

The polynomial \(x^4 + 3x^2 + 2\) factors as \((x^2 + 1)(x^2 + 2)\).

Step 1: Identify Possible Factorization

To factor the polynomial \(x^4 + 3x^2 + 2\), first observe that it is a quadratic in form where \(y = x^2\). This implies the polynomial can be expressed as \(y^2 + 3y + 2\). We'll attempt to factor this auxiliary quadratic expression.

Step 2: Factor the Quadratic Expression

The quadratic \(y^2 + 3y + 2\) can be factored by finding two numbers that multiply to \(2\) (the constant term) and add to \(3\) (the coefficient of \(y\)). These two numbers are \(1\) and \(2\). Thus, \(y^2 + 3y + 2 = (y + 1)(y + 2)\).

Step 3: Substitute Back to the Original Variable

Substitute \(y = x^2\) back into the factored form to write the polynomial as a product of polynomials in terms of \(x\). This gives \((x^2 + 1)(x^2 + 2)\).

Step 4: Confirm Irreducibility Over Real Numbers

Verify that the polynomials \(x^2 + 1\) and \(x^2 + 2\) cannot be further factored into polynomials with real coefficients of degree less than two. Both are irreducible over the real numbers because they have no real roots.

Key Concepts

Irreducible Polynomials

Understanding irreducible polynomials can simplify a complex polynomial into smaller, simpler components that can't be factored further using coefficients from a given field, in this case, the real numbers. An irreducible polynomial is a polynomial that cannot be factored into polynomials of lower degree with coefficients in the same field. This is much like trying to break down a prime number into factors; it just can't be done further within the set of integers.
This mathematical property is widely used to simplify expressions and solve equations.

• In our example, the polynomials \(x^2 + 1\) and \(x^2 + 2\) are irreducible over the real numbers because they do not have any real roots.
• Attempting to solve \(x^2 + 1 = 0\) or \(x^2 + 2 = 0\) would involve taking square roots of negative numbers, resulting in complex numbers, not real ones.

So, these factors, \(x^2 + 1\) and \(x^2 + 2\), are the simplest form accessible for our given purpose with real numbers.

Quadratic Form

When dealing with polynomials, identifying if a polynomial can be expressed in a quadratic form greatly aids in simplifying the factorization process. A polynomial is said to be in quadratic form if it can be expressed as \(ay^2 + by + c\), where the term \(y\) itself is a function of another variable.
For example, consider \(x^4 + 3x^2 + 2\). This polynomial can be transformed into a quadratic form by considering a substitution, like \(y = x^2\). Our original polynomial then becomes \(y^2 + 3y + 2\), which is a recognizable quadratic equation.

• This approach effectively reduces the problem complexity, allowing for easier factorization using standard methods for quadratic expressions.
• Once the quadratic is factored, substituting back the original variable enables us to express the polynomial using the actual variable terms, without altering the problem's nature.

This method of utilizing quadratic forms helps in tackling more complex polynomial equations in a straightforward manner.

Real Coefficients

When working with real numbers, real coefficients are key to understanding polynomials' behavior. Real coefficients are simply coefficients that are not imaginary or complex numbers. They make up every term in a real polynomial, which are polynomials that, upon being given real inputs, return real outputs.
Irreducibility over the real numbers is determined by these coefficients because it is beneficial to know whether further simplification with real coefficients is possible.

• This simplification helps determine if complex or imaginary numbers are needed for factorization or solving.
• For the factors \(x^2 + 1\) and \(x^2 + 2\) from the original polynomial, both have real coefficients, making them convenient candidates for irreducibility checking over the real numbers.

Therefore, the lack of real roots for these polynomials implies they remain irreducible in the realm of real numbers. This insight helps in solutions where only real coefficients are permitted or preferred.