/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 72 Perform each division. $$ \f... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 72

Perform each division. $$ \frac{y^{3}-1}{y-1} $$

Short Answer

Expert verified
The simplified form is \(y^2 + y + 1\).

Step by step solution

01

- Recognize the division

The task is to divide \(\frac{y^3 - 1}{y - 1}\). This polynomial division can be simplified using algebraic identities.
02

- Identify the algebraic identity

Notice that \(y^3 - 1\) can be expressed using the difference of cubes identity: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Here, \(a = y\) and \(b = 1\).
03

- Apply the algebraic identity

Rewrite \(y^3 - 1\) as \( (y - 1)(y^2 + y \cdot 1 + 1^2) \). This simplifies to: \((y - 1)(y^2 + y + 1)\).
04

- Simplify the division

Now perform the division: \(\frac{(y - 1)(y^2 + y + 1)}{y - 1} \). The \(y - 1\) terms cancel out, leaving \(y^2 + y + 1\).

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Key Concepts

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

Algebraic Identities
Algebraic identities are powerful tools that simplify polynomial operations.
They allow us to rewrite complex expressions in simpler forms.
These identities include well-known formulas such as the difference of squares, cube identities, and more.
In our division problem, we used the difference of cubes identity.
This identity states: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Recognizing and applying these identities can make polynomial division much easier.
Here, we saw that \(y^3 - 1\) was rewritten using the identity, which greatly simplified our task.
Difference of Cubes
The difference of cubes is a specific algebraic identity useful for simplifying certain polynomial expressions.
It deals with expressions where two cube terms are subtracted, like \(a^3 - b^3\).
The formula for difference of cubes is: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
In our example, \(a = y\) and \(b = 1\).
Applying the identity, \(y^3 - 1\) becomes \((y - 1)(y^2 + y + 1)\).
This transformation allowed us to factor the polynomial and make the division straightforward.
Understanding this formula is crucial for similar problems involving cubes.
Simplifying Polynomials
Simplifying polynomials involves breaking down complex expressions into simpler forms. It often requires recognizing and applying algebraic identities.
For example, in our division problem, we began with \(y^3 - 1\).
Using the difference of cubes identity, we rewrote it as \((y - 1)(y^2 + y + 1)\).
This step greatly simplified the division.
Finally, we canceled out \(y - 1\) from the numerator and denominator, leaving us with the simple result \(y^2 + y + 1\).
Recognizing patterns and applying appropriate identities are key strategies in polynomial simplification.

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Most popular questions from this chapter

Simplify by writing each expression with positive exponents. Assume that all variables represent nonzero real numbers. See Example \(5 .\) $$ \frac{\left(m^{8} n^{-4}\right)^{2}}{m^{-2} n^{5}} $$

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