Question

f(x)=x⁴+2x²-8, Write the polynomial as the product of factors that are irreducible over the rationals.

Answer 1

To write the polynomial as the product of factors that are irreducible over the rational, we need to factorize it. The polynomial f(x) can be written as x²(x + 2)(x - 2).

Step-by-step explanation:

To write the polynomial as the product of factors that are irreducible over the rational, we need to factorize it. Let's factorize it step by step:

Step 1: Factor out the common factor x² from the polynomial.

f(x) = x²(x² + 2 - 8/x²)

Step 2: Factor the quadratic expression inside the parentheses.

f(x) = x²(x + 2)(x - 2)

So, the polynomial f(x) can be written as the product of factors that are irreducible over the rational as x²(x + 2)(x - 2).

Answer 2

`\[ f(x) = (x^2 + 2)(x^2 - 2) \]`

Step-by-step explanation:

The given polynomial
`\( f(x) = x^4 + 2x^2 - 8 \)` can be factored into irreducible factors over the rationals as
`\( (x^2 + 2)(x^2 - 2) \).`

To understand this factorization, let's break down the process. Start by factoring out any common factors. In this case, there are no common factors other than 1.

The polynomial is a quartic expression, so we can proceed by factoring it into two quadratic factors.

We notice that
`\( x^2 + 2 \)` is the sum of squares, and
`\( x^2 - 2 \)` is the difference of squares. The sum of squares cannot be factored further over the rationals, but the difference of squares can be factored into
`\( (x + √(2))(x - √(2)) \).`

Therefore, the final factorization is
`\( (x^2 + 2)(x^2 - 2) \),` where both factors are irreducible over the rationals. This means that the quadratic factors cannot be factored further into linear factors with rational coefficients.

The irreducible factors over the rationals are
`\( x^2 + 2 \) and \( x^2 - 2 \),` and their product gives the original polynomial.