Math Problem Statement

Solution

The problem now asks to rewrite the expression α4+β4αβ\frac{\alpha^4 + \beta^4}{\alpha \beta} using (α+β)(\alpha + \beta) and (αβ)(\alpha \beta).

Let's start by expressing α4+β4\alpha^4 + \beta^4 in terms of (α+β)(\alpha + \beta) and (αβ)(\alpha \beta).

Step-by-Step Solution:

  1. Square the sum of squares:
    We can express α4+β4\alpha^4 + \beta^4 as (α2+β2)22(αβ)2(\alpha^2 + \beta^2)^2 - 2(\alpha \beta)^2.

  2. Express α2+β2\alpha^2 + \beta^2 using (α+β)(\alpha + \beta) and (αβ)(\alpha \beta):
    The identity α2+β2=(α+β)22αβ\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta can be used.

    So, α2+β2=(α+β)22αβ\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta

  3. Substitute back into the expression for α4+β4\alpha^4 + \beta^4:
    α4+β4=((α+β)22αβ)22(αβ)2\alpha^4 + \beta^4 = ((\alpha + \beta)^2 - 2\alpha \beta)^2 - 2(\alpha \beta)^2

  4. Expand and simplify:
    (α+β)44(α+β)2(αβ)+4(αβ)22(αβ)2(\alpha + \beta)^4 - 4(\alpha + \beta)^2(\alpha \beta) + 4(\alpha \beta)^2 - 2(\alpha \beta)^2

    Combine like terms: α4+β4=(α+β)44(α+β)2(αβ)+2(αβ)2\alpha^4 + \beta^4 = (\alpha + \beta)^4 - 4(\alpha + \beta)^2(\alpha \beta) + 2(\alpha \beta)^2

  5. Rewrite the original fraction:
    α4+β4αβ=(α+β)44(α+β)2(αβ)+2(αβ)2αβ\frac{\alpha^4 + \beta^4}{\alpha \beta} = \frac{(\alpha + \beta)^4 - 4(\alpha + \beta)^2(\alpha \beta) + 2(\alpha \beta)^2}{\alpha \beta}

  6. Divide each term by αβ\alpha \beta:
    (α+β)4αβ4(α+β)2(αβ)αβ+2(αβ)2αβ\frac{(\alpha + \beta)^4}{\alpha \beta} - 4\frac{(\alpha + \beta)^2(\alpha \beta)}{\alpha \beta} + 2\frac{(\alpha \beta)^2}{\alpha \beta}

    Simplifying each term: =(α+β)4αβ4(α+β)2+2αβ= \frac{(\alpha + \beta)^4}{\alpha \beta} - 4(\alpha + \beta)^2 + 2\alpha \beta

Final Expression:

Thus, the expression α4+β4αβ\frac{\alpha^4 + \beta^4}{\alpha \beta} written in terms of (α+β)(\alpha + \beta) and (αβ)(\alpha \beta) is:

(α+β)4αβ4(α+β)2+2αβ\frac{(\alpha + \beta)^4}{\alpha \beta} - 4(\alpha + \beta)^2 + 2\alpha \beta

Would you like more details or have any questions?

Here are some questions to further explore this topic:

  1. How would you generalize the expression for αn+βnαβ\frac{\alpha^n + \beta^n}{\alpha \beta} for any nn?
  2. What happens to the expression if α=β\alpha = \beta?
  3. Can you derive a similar expression for α5+β5\alpha^5 + \beta^5 using (α+β)(\alpha + \beta) and (αβ)(\alpha \beta)?
  4. What other polynomial identities can be helpful in problems like this?
  5. How can these identities help in solving polynomial equations?

Tip: Always look for patterns in polynomial expressions. Understanding these patterns can simplify your work with algebraic identities.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Polynomial Identities
Quadratic Equations

Formulas

Sum of powers of roots formula

Theorems

Binomial theorem

Suitable Grade Level

Grades 11-12