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,...
Related questions
Question
![### Lesson on Factoring Polynomial Expressions
**Objective:**
Learn how to completely factor polynomial expressions, including factoring out common terms and recognizing when to factor in a negative one (-1).
#### Example Problem:
Completely factor the expression (including -1, if necessary):
\[ -18x^4y^3z^2 + 27x^3y^2z - 21x^2y \]
**Steps to Solve:**
1. **Identify Common Factors:**
Look for common factors in each term of the polynomial. In this case, observe that each term contains \( x^2 y \).
2. **Factor Out the GCF (Greatest Common Factor):**
The GCF here is \( 3x^2 y \). Factoring \( 3x^2 y \) out of each term, we get:
\[ -18x^4y^3z^2 = 3x^2y(-6x^2yz^2) \]
\[ 27x^3y^2z = 3x^2y(9xyz) \]
\[ -21x^2y = 3x^2y(-7) \]
Combining these, we have:
\[ -18x^4y^3z^2 + 27x^3y^2z - 21x^2y = 3x^2 y (-6x^2 yz^2 + 9xyz - 7) \]
3. **Check for Additional Factoring:**
Check if the polynomial inside the parentheses can be factored further. The expression \( -6x^2 yz^2 + 9xyz - 7 \) doesn’t factor further in a simple manner.
Therefore, the completely factored form of the expression is:
\[ 3x^2 y (-6x^2 yz^2 + 9xyz - 7) \]
**Feedback:**
The incorrect attempt:
\[ 3x^2y(-6x^2 yz^2 + 9xyz - 7) \]
**Explanation:**
The calculation successfully factored out the greatest common factor but did not find any further simplification within the inner polynomial. The correct manner of representation matches the structure but, depending on the inputs, factoring -1 may be necessary if the sign of the leading term was positive initially and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F758f6220-2c9c-498c-a499-60dbbe8461af%2F6b2eb884-5b4c-4d68-b96a-405eb07f21f6%2Fmjm2zox_processed.png&w=3840&q=75)
Transcribed Image Text:### Lesson on Factoring Polynomial Expressions
**Objective:**
Learn how to completely factor polynomial expressions, including factoring out common terms and recognizing when to factor in a negative one (-1).
#### Example Problem:
Completely factor the expression (including -1, if necessary):
\[ -18x^4y^3z^2 + 27x^3y^2z - 21x^2y \]
**Steps to Solve:**
1. **Identify Common Factors:**
Look for common factors in each term of the polynomial. In this case, observe that each term contains \( x^2 y \).
2. **Factor Out the GCF (Greatest Common Factor):**
The GCF here is \( 3x^2 y \). Factoring \( 3x^2 y \) out of each term, we get:
\[ -18x^4y^3z^2 = 3x^2y(-6x^2yz^2) \]
\[ 27x^3y^2z = 3x^2y(9xyz) \]
\[ -21x^2y = 3x^2y(-7) \]
Combining these, we have:
\[ -18x^4y^3z^2 + 27x^3y^2z - 21x^2y = 3x^2 y (-6x^2 yz^2 + 9xyz - 7) \]
3. **Check for Additional Factoring:**
Check if the polynomial inside the parentheses can be factored further. The expression \( -6x^2 yz^2 + 9xyz - 7 \) doesn’t factor further in a simple manner.
Therefore, the completely factored form of the expression is:
\[ 3x^2 y (-6x^2 yz^2 + 9xyz - 7) \]
**Feedback:**
The incorrect attempt:
\[ 3x^2y(-6x^2 yz^2 + 9xyz - 7) \]
**Explanation:**
The calculation successfully factored out the greatest common factor but did not find any further simplification within the inner polynomial. The correct manner of representation matches the structure but, depending on the inputs, factoring -1 may be necessary if the sign of the leading term was positive initially and
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education