Problem 102 Simplify. $$ \frac{x^{3}+x^{... [FREE SOLUTION]

Chapter 7: Problem 102

Simplify. $$ \frac{x^{3}+x^{2}-y^{3}-y^{2}}{x^{2}-2 x y+y^{2}} $$

Short Answer

Expert verified

\[\frac{x^{2}(x+1) - y^{2}(y+1)}{(x - y)^{2}}\]

Step by step solution

Factor the Numerator

First, factor the numerator. Observe that both the terms involving x and y can be grouped:\[ x^{3}+x^{2}-y^{3}-y^{2} = x^{2}(x+1) - y^{2}(y+1)\]

Factor the Denominator

Next, factor the denominator. Recognize the quadratic as a perfect square:\[ x^{2} - 2xy + y^{2} = (x - y)^{2}\]

Combine and Simplify

Combine the factored forms and simplify:\[\frac{x^{2}(x+1) - y^{2}(y+1)}{(x - y)^{2}}\]Notice that the numerator does not factor further in a way that would simplify with the denominator. However, you can use polynomial division if necessary for specific values or variable comforts.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring

Factoring is like breaking down a number or an expression into its components. For example, the number 6 can be factored into 2 and 3 because 2 * 3 = 6. Similarly, in algebra, factoring an expression means writing it as a product of simpler expressions.

In the given exercise, the numerator $x^3 + x^2 - y^3 - y^2$ is factored by grouping. Here鈥檚 the process:

Group the terms involving $x$: $x^3 + x^2$
Group the terms involving $y$: $-y^3 - y^2$
Factor each group: $x^2(x + 1)$ and $-y^2(y + 1)$
Combine them to form the factored numerator: $x^2(x + 1) - y^2(y + 1)$.

This step-by-step approach helps break down complex polynomials into simpler factors.

Quadratic Equations

Quadratic equations are polynomials of degree 2. The standard form is $ax^2 + bx + c = 0$. When you see an equation like $x^2 - 2xy + y^2$, it may look complex, but it鈥檚 still a quadratic in disguise.

In this exercise, the denominator is $x^2 - 2xy + y^2$. Recognize that this is a perfect square trinomial. It can be factored into $(x - y)^2$ because:

Expanding $(x - y)^2$ gives $x^2 - 2xy + y^2$
This process makes the original expression simpler.

Understanding quadratics is crucial in simplifying polynomial expressions and solving algebraic equations.

Polynomial Division

Polynomial division is a method used if the numerator and denominator do not simplify directly through factoring. To divide polynomials, follow these steps:

Divide the first term of the numerator by the first term of the denominator
Multiply the entire divisor by this result and subtract from the original numerator
Repeat this process with the resulting polynomial until the degree is less than that of the divisor.

Although polynomial division wasn鈥檛 necessary in this exercise, knowing it gives you another tool for simplification. It鈥檚 useful when factorizing doesn鈥檛 lead to an immediate simplification. Practice these steps to master polynomial division, adding depth to your algebraic skills.

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91影视

Simplify. $$ \frac{x^{3}+x^{2}-y^{3}-y^{2}}{x^{2}-2 x y+y^{2}} $$

Short Answer

Step by step solution

Factor the Numerator

Factor the Denominator

Combine and Simplify

Key Concepts

Factoring

Quadratic Equations

Polynomial Division

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