Simplifying $\frac{a-a^4}{a^2+a+1}$: A Step-by-Step Guide

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In this article, we'll break down how to simplify the algebraic expression $\frac{a-a^4}{a^2+a+1}$. This involves factoring, identifying common terms, and then simplifying the expression to its most reduced form. Understanding these steps is crucial for anyone studying algebra, as it showcases fundamental techniques used in various mathematical contexts. Let's dive in and make this process clear and straightforward!

1. Understanding the Expression

Before we begin simplifying, let's take a closer look at the expression we're dealing with: $\frac{a-a^4}{a^2+a+1}$.

• Numerator: The numerator is $a - a^4$. Notice that both terms have 'a' as a common factor. Factoring out 'a' is our first step.
• Denominator: The denominator is $a^2 + a + 1$. This quadratic expression doesn't immediately factor using simple methods, but we'll keep it in mind as we proceed.

Our main goal is to simplify this fraction by factoring both the numerator and the denominator, and then canceling out any common factors. This process makes the expression cleaner and easier to work with in further calculations. So, let's get started with factoring the numerator.

2. Factoring the Numerator: $a - a^4$

The key to simplifying algebraic expressions often lies in factoring. Factoring allows us to break down complex terms into simpler components, revealing common factors that can be canceled out. In the case of the numerator, $a - a^4$, the first step is to identify the greatest common factor (GCF). Here, the GCF is 'a'.

Step-by-Step Factoring

1. Identify the GCF: As mentioned, the greatest common factor of $a$ and $a^4$ is $a$.
2. Factor out the GCF:
   • $a - a^4 = a(1 - a^3)$

Now, we have $a(1 - a^3)$. But we're not done yet! The term $(1 - a^3)$ is a difference of cubes, which has a special factoring pattern. Recognizing and applying this pattern is crucial for further simplification.

Difference of Cubes

The difference of cubes pattern is: $A^3 - B^3 = (A - B)(A^2 + AB + B^2)$.

In our case, we have $1 - a^3$, which can be seen as $1^3 - a^3$. Here:

• $A = 1$
• $B = a$

Applying the pattern:

$1 - a^3 = (1 - a)(1^2 + 1*a + a^2) = (1 - a)(1 + a + a^2)$

Complete Factoring of the Numerator

Putting it all together, the factored form of the numerator $a - a^4$ is:

$a - a^4 = a(1 - a^3) = a(1 - a)(1 + a + a^2)$

Now that we've successfully factored the numerator, we're one step closer to simplifying the original expression. Next, we'll revisit the denominator and see if it has any common factors with the factored numerator. This careful, step-by-step approach ensures we don't miss any simplification opportunities.

3. Revisiting the Denominator: $a^2 + a + 1$

After factoring the numerator, our next focus is the denominator, $a^2 + a + 1$. This quadratic expression is a bit trickier to factor than the terms we dealt with in the numerator. Often, the key to simplifying complex fractions is finding common factors between the numerator and the denominator. So, we need to consider: can $a^2 + a + 1$ be factored further, and does it share any factors with the factored form of the numerator, which is $a(1 - a)(1 + a + a^2)$?

Checking for Factorability

To determine if $a^2 + a + 1$ can be factored using elementary methods, we can try to find two numbers that multiply to 1 (the constant term) and add up to 1 (the coefficient of the 'a' term). However, there are no such integers. This suggests that $a^2 + a + 1$ may not factor neatly using simple integer coefficients.

Recognizing a Key Term

Looking back at the factored numerator, we see a term that looks very similar: $(1 + a + a^2)$. This is the same as $a^2 + a + 1$, just written in a different order. Recognizing this similarity is crucial because it points to a significant simplification opportunity.

The Significance of Matching Terms

The fact that the denominator $a^2 + a + 1$ exactly matches the quadratic term in the factored numerator is not a coincidence. It's a deliberate part of the problem design, meant to lead us towards canceling out these terms. This is a common technique in simplifying algebraic expressions: identifying and canceling common factors to reduce the expression to its simplest form.

Now that we've confirmed the match, we're ready to put everything together and simplify the entire expression. The next step involves writing out the fraction with the factored numerator and the denominator, and then performing the cancellation.

4. Simplifying the Expression: Putting It All Together

Now we've reached the heart of the problem: simplifying the expression using the factored forms we've found. This involves rewriting the original expression with the factored numerator and then looking for common terms to cancel out.

Writing the Factored Expression

Recall the original expression:

$\frac{a-a^4}{a^2+a+1}$

We factored the numerator as:

$a - a^4 = a(1 - a)(1 + a + a^2)$

And we recognized that the denominator is:

$a^2 + a + 1$

Which is the same as the quadratic term in the factored numerator, $(1 + a + a^2)$.

So, we can rewrite the original expression as:

$\frac{a(1 - a)(1 + a + a^2)}{a^2 + a + 1}$

Canceling Common Factors

Now, we can clearly see that the term $(1 + a + a^2)$ in the numerator is identical to the denominator $(a^2 + a + 1)$. This allows us to cancel these common factors:

$\frac{a(1 - a)(1 + a + a^2)}{a^2 + a + 1} = \frac{a(1 - a)\cancel{(1 + a + a^2)}}{\cancel{(a^2 + a + 1)}}$

After canceling the common terms, we are left with:

$a(1 - a)$

Simplified Expression

Finally, we can distribute the 'a' to get the simplified expression:

$a(1 - a) = a - a^2$

Therefore, the simplified form of the original expression $\frac{a-a^4}{a^2+a+1}$ is $a - a^2$. This process highlights the power of factoring in simplifying algebraic expressions, allowing us to reduce complex fractions to more manageable forms.

5. Final Answer and Conclusion

After meticulously working through the steps of factoring and simplifying, we've arrived at the final answer. The original expression, $\frac{a-a^4}{a^2+a+1}$, simplifies to:

$a - a^2$

This result underscores the importance of mastering factoring techniques in algebra. By breaking down complex expressions into their factors, we can identify common terms that can be canceled out, leading to a much simpler form. In this case, we factored the numerator, recognized a key quadratic expression, and then simplified by canceling matching terms.

Key Takeaways

• Factoring is Crucial: Factoring is a fundamental skill in algebra. It allows you to simplify complex expressions and solve equations.
• Recognize Patterns: Identifying patterns like the difference of cubes ($A^3 - B^3$) can significantly speed up the simplification process.
• Look for Common Factors: Always look for common factors between the numerator and the denominator in a fraction. Canceling these factors is key to simplifying the expression.
• Step-by-Step Approach: Break down the problem into smaller, manageable steps. This makes the process less daunting and reduces the chance of errors.

By following these steps and practicing regularly, you can confidently tackle similar simplification problems. Algebra is a building block for more advanced mathematics, and mastering these fundamental skills is essential for success.

For further learning and practice on algebraic simplification, you may find helpful resources at Khan Academy Algebra. This site offers lessons, practice exercises, and videos that can help you strengthen your algebra skills.