Problem 56 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. $$f(x)=x^{4}+6 x^{2}-27$$

Step-by-Step Solution

Answer

The polynomial \(x^{4}+6 x^{2}-27\) can be factored over the rationals as \((x^{2}-3) (x^{2}+9)\), over the reals as \((x^{2}-3) (x^{2}+9)\), and as completely factored using complex numbers as \((x^{2}-3) (x+3i)(x-3i)\).

Step 1: Factor the polynomial over the rationals

We start by recognizing this as a quadratic in \(x^2\). We can rewrite the polynomial as \(x^{4}+6 x^{2}-27 = (x^{2})^{2}+6 x^{2}-27\), which can be factored by grouping as: \((x^{2})^{2}+6 x^{2}-27 = (x^{2}-3) (x^{2}+9)\)

Step 2: Factor the polynomial over the reals

Next, we factorize the equation over the reals by identifying the parts that can be further decomposed into linear factors, quadratic factors or remain unchanged. In this case, all terms are quadratic so they remain as is. So, the polynomial remains as \((x^{2}-3) (x^{2}+9)\)

Step 3: Factor the polynomial completely

Finally, we further decompose it into its complex factors. We note that the term \(x^{2}+9\) has complex factors since the number inside the square root is negative. We can write \(x^{2}+9\) as \((x+3i)(x-3i)\). Hence, the polynomial in fully factored form is \((x^{2}-3) (x+3i)(x-3i)\).