Unit 7: Polynomials and Factoring - Homework 1 Answer Key: A Comprehensive Guide
This guide provides comprehensive answers and explanations for Homework 1 in Unit 7, focusing on polynomials and factoring. Since I don't have access to your specific homework assignment, I will provide a general framework covering common problems encountered in this unit. This will allow you to check your work and understand the underlying concepts. Remember to always refer to your textbook and class notes for specific problem details and terminology.
Understanding Polynomials and Factoring
Before diving into the answers, let's review the key concepts:
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Polynomials: These are algebraic expressions with multiple terms, each consisting of a constant multiplied by a variable raised to a non-negative integer power. Examples include 3x² + 2x - 5 and 7y⁴ - 1.
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Factoring: This is the process of breaking down a polynomial into simpler expressions that, when multiplied, give the original polynomial. Factoring is crucial for solving polynomial equations and simplifying expressions. Common factoring techniques include:
- Greatest Common Factor (GCF): Finding the largest factor common to all terms.
- Difference of Squares: Factoring expressions of the form a² - b² into (a + b)(a - b).
- Trinomial Factoring: Factoring quadratic expressions of the form ax² + bx + c.
- Grouping: Factoring polynomials with four or more terms by grouping terms with common factors.
Example Problems and Solutions
Let's tackle some typical problems found in Homework 1 of Unit 7. Remember to replace these examples with your actual homework problems.
Problem 1: Find the GCF of 12x³ and 18x²
Solution:
- Find the GCF of the coefficients: The GCF of 12 and 18 is 6.
- Find the GCF of the variables: The GCF of x³ and x² is x².
- The GCF of 12x³ and 18x² is 6x².
Problem 2: Factor the expression x² - 25
Solution:
This is a difference of squares (a² - b², where a = x and b = 5). Therefore, the factored form is (x + 5)(x - 5).
Problem 3: Factor the trinomial 3x² + 7x + 2
Solution:
We need to find two numbers that add up to 7 (the coefficient of x) and multiply to 6 (the product of 3 and 2). These numbers are 6 and 1. We rewrite the middle term:
3x² + 6x + x + 2
Now we factor by grouping:
3x(x + 2) + 1(x + 2)
This simplifies to (3x + 1)(x + 2).
Problem 4: Factor the expression 4x³ + 8x² - 12x
Solution:
First, find the GCF of all terms, which is 4x:
4x(x² + 2x - 3)
Now factor the quadratic trinomial:
4x(x + 3)(x - 1)
Addressing Specific Homework Questions:
To get more specific help, please provide the actual problems from your Homework 1. I can then offer detailed solutions and explanations tailored to your assignment. This approach ensures you understand the concepts and can solve similar problems independently.
Remember to always show your work, clearly explaining each step of the factoring process. This not only helps you understand the material better but also allows your teacher to see your thought process and provide more targeted feedback.
