Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![### Proving the Polynomial Identity for the Cube of a Binomial Representing a Difference
This document demonstrates the mathematical proof for the cube of a binomial representing a difference, specifically for \((x - y)^3\).
Consider the binomial \((x - y)\). To prove the polynomial identity for its cube, we have:
\[ (x - y)^3 \]
### Step-by-Step Expansion
1. **Initial Expression:**
\[
(x - y)^3
\]
2. **Applying the Formula for Cubing a Binomial:**
\[
(x - y)^3 = (x - y) \cdot (x - y)^2
\]
3. **Expanding the Square:**
\[
(x - y)^2 = x^2 - 2xy + y^2
\]
So,
\[
(x - y)^3 = (x - y) \cdot (x^2 - 2xy + y^2)
\]
4. **Distributing the Binomial:**
\[
(x - y) \cdot (x^2 - 2xy + y^2) = x(x^2 - 2xy + y^2) - y(x^2 - 2xy + y^2)
\]
5. **Further Expansion:**
\[
x(x^2 - 2xy + y^2) = x^3 - 2x^2y + xy^2
\]
\[
y(x^2 - 2xy + y^2) = yx^2 - 2xy^2 + y^3
\]
Combining these results:
\[
x^3 - 2x^2y + xy^2 - (yx^2 - 2xy^2 + y^3)
\]
6. **Simplifying the Expression:**
\[
= x^3 - 2x^2y + xy^2 - yx^2 + 2xy^2 - y^3
\]
Combine like terms:
\[
= x^3 - 3x^2y + 3xy^2 - y^3
\]
### Final Polynomial Identity
Therefore, the polynomial identity for](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2a18a320-2046-4089-b27e-0ca38db1d2b2%2Fa31bddc7-3800-42d5-ac45-99bd5f9d1d17%2Fr71jt11h.jpeg&w=3840&q=75)
Transcribed Image Text:### Proving the Polynomial Identity for the Cube of a Binomial Representing a Difference
This document demonstrates the mathematical proof for the cube of a binomial representing a difference, specifically for \((x - y)^3\).
Consider the binomial \((x - y)\). To prove the polynomial identity for its cube, we have:
\[ (x - y)^3 \]
### Step-by-Step Expansion
1. **Initial Expression:**
\[
(x - y)^3
\]
2. **Applying the Formula for Cubing a Binomial:**
\[
(x - y)^3 = (x - y) \cdot (x - y)^2
\]
3. **Expanding the Square:**
\[
(x - y)^2 = x^2 - 2xy + y^2
\]
So,
\[
(x - y)^3 = (x - y) \cdot (x^2 - 2xy + y^2)
\]
4. **Distributing the Binomial:**
\[
(x - y) \cdot (x^2 - 2xy + y^2) = x(x^2 - 2xy + y^2) - y(x^2 - 2xy + y^2)
\]
5. **Further Expansion:**
\[
x(x^2 - 2xy + y^2) = x^3 - 2x^2y + xy^2
\]
\[
y(x^2 - 2xy + y^2) = yx^2 - 2xy^2 + y^3
\]
Combining these results:
\[
x^3 - 2x^2y + xy^2 - (yx^2 - 2xy^2 + y^3)
\]
6. **Simplifying the Expression:**
\[
= x^3 - 2x^2y + xy^2 - yx^2 + 2xy^2 - y^3
\]
Combine like terms:
\[
= x^3 - 3x^2y + 3xy^2 - y^3
\]
### Final Polynomial Identity
Therefore, the polynomial identity for
![### Polynomials Multiplied Together
Let's explore the polynomial multiplication process step-by-step. Below is a detailed explanation.
---
#### Step-by-Step Explanation:
1. **First Task**:
- Multiply \((x - y)\) with each term in the polynomial \[(x^2 - 2xy + y^2)\).
2. **Second Task**:
- Multiply these results with \((x - y)\), then sum all these products.
3. **Final Expression**:
- Add up all the products to get the final polynomial.
---
### Tables to Match the Steps:
1. **Polynomial Terms**:
\[
\begin{array}{|c|c|}
\hline
\text{Terms} & \text{Values} \\
\hline
(x^2 - 2xy + y^2) & x^2 - 2xy + y^2 \\
\hline
(x - y) & x - y \\
\hline
\end{array}
\]
2. **Multiplication Steps**:
\[
\begin{array}{|c|c|}
\hline
(x - y)(x - y) & x^2 - 2xy + y^2 \\
\hline
(x - y)(x^2 - 2xy + y^2) & x^3 - 3x^2y + 3xy^2 - y^3 \\
\hline
\end{array}
\]
3. **Intermediary Steps in Detail**:
\[
\begin{array}{|c|c|c|}
\hline
\text{Expression} & \text{Step} & \text{Result} \\
\hline
x^3 - 3x^2y & Step 1 & x^3 - 3x^2y \\
\hline
+ 3xy^2 - y^3 & Step 2 & + 3xy^2 - y^3 \\
\hline
= x^3 - 3x^2y + 3xy^2 - y^3 & Final Step & x^3 - 3x^2y +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2a18a320-2046-4089-b27e-0ca38db1d2b2%2Fa31bddc7-3800-42d5-ac45-99bd5f9d1d17%2F33p1lnn.jpeg&w=3840&q=75)
Transcribed Image Text:### Polynomials Multiplied Together
Let's explore the polynomial multiplication process step-by-step. Below is a detailed explanation.
---
#### Step-by-Step Explanation:
1. **First Task**:
- Multiply \((x - y)\) with each term in the polynomial \[(x^2 - 2xy + y^2)\).
2. **Second Task**:
- Multiply these results with \((x - y)\), then sum all these products.
3. **Final Expression**:
- Add up all the products to get the final polynomial.
---
### Tables to Match the Steps:
1. **Polynomial Terms**:
\[
\begin{array}{|c|c|}
\hline
\text{Terms} & \text{Values} \\
\hline
(x^2 - 2xy + y^2) & x^2 - 2xy + y^2 \\
\hline
(x - y) & x - y \\
\hline
\end{array}
\]
2. **Multiplication Steps**:
\[
\begin{array}{|c|c|}
\hline
(x - y)(x - y) & x^2 - 2xy + y^2 \\
\hline
(x - y)(x^2 - 2xy + y^2) & x^3 - 3x^2y + 3xy^2 - y^3 \\
\hline
\end{array}
\]
3. **Intermediary Steps in Detail**:
\[
\begin{array}{|c|c|c|}
\hline
\text{Expression} & \text{Step} & \text{Result} \\
\hline
x^3 - 3x^2y & Step 1 & x^3 - 3x^2y \\
\hline
+ 3xy^2 - y^3 & Step 2 & + 3xy^2 - y^3 \\
\hline
= x^3 - 3x^2y + 3xy^2 - y^3 & Final Step & x^3 - 3x^2y +
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