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Magazine Chapter 5: Problem 58
Divide. $$ \frac{x^{6}-x^{4}}{x^{3}+1} $$
Short Answer
Expert verified
\(\frac{x^4(x - 1)}{x^2 - x + 1}\).
Step by step solution
01
Simplifying the Dividend
First, factor out the greatest common factor in the numerator, which is \(x^4\). This simplifies the expression to \(x^4(x^2 - 1)\). Then, further factor \(x^2 - 1\) as it is a difference of squares. This gives us \(x^4(x + 1)(x - 1)\).
02
Rewriting the Expression
Now that the numerator is rewritten, the expression becomes \(\frac{x^4(x + 1)(x - 1)}{x^3 + 1}\).
03
Factoring the Denominator
Factor the denominator \(x^3 + 1\). This can be done using the sum of cubes formula: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Here, \(a = x\) and \(b = 1\), so \(x^3 + 1\) factors as \((x + 1)(x^2 - x + 1)\).
04
Canceling Common Factors
In the expression \(\frac{x^4(x + 1)(x - 1)}{(x + 1)(x^2 - x + 1)}\), we can cancel the common factor \((x + 1)\) from the numerator and the denominator. This simplifies the division to \(\frac{x^4(x - 1)}{x^2 - x + 1}\).
05
Final Expression
The simplified form of the original expression \(\frac{x^6-x^4}{x^3+1}\) is \(\frac{x^4(x - 1)}{x^2 - x + 1}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factoring Polynomials
Factoring polynomials is the process of breaking down a polynomial into simpler terms, or "factors," that are multiplied together to give the original polynomial. This is similar to factoring numbers, like breaking down 12 into 3 and 4. By finding the factors of a polynomial, you can simplify expressions and solve equations more easily.
- In the context of the problem, the numerator of the original expression \( x^6 - x^4 \) can be factored by identifying a common factor in all terms.
- The greatest common factor for the terms is \( x^4 \), which simplifies \( x^6 - x^4 \) to \( x^4(x^2 - 1) \).
- We then further factor the expression \( x^2 - 1 \) using the difference of squares method, which breaks it into \( (x + 1)(x - 1) \).
Greatest Common Factor
The greatest common factor (GCF) is the highest degree of a variable or the largest number that divides two or more terms without leaving a remainder. It is a key concept when simplifying expressions and solving equations in algebra.
- In polynomials, the GCF is identified by looking at each term and determining which factor or factors are common. In this problem, \( x^6 \) and \( x^4 \) both have the factor \( x^4 \) in common.
- This means we can factor \( x^4 \) out of both terms, revising \( x^6 - x^4 \) to \( x^4(x^2 - 1) \).
Sum of Cubes
The sum of cubes refers to a polynomial expression of the type \( a^3 + b^3 \). It can be factored using a special formula, which is critical when simplifying polynomials or dividing rational expressions.
- The sum of cubes formula is \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \). This formula helps break down complex expressions into simpler factors for easier manipulation.
- In the solved exercise, the denominator \( x^3 + 1 \) can be seen as a sum of cubes, where \( a = x \) and \( b = 1 \). Thus, we can factor it as \( (x + 1)(x^2 - x + 1) \).
