Unit 7 Polynomials And Factoring Homework 7 Answer Key

Unit 7 Polynomials and Factoring: Homework 7 Answer Key & Comprehensive Guide

This comprehensive guide tackles Unit 7's polynomials and factoring, specifically focusing on Homework 7. We'll delve into the core concepts, provide solutions, and offer strategies to master this crucial algebra topic. Remember, understanding the why behind the answers is far more valuable than simply memorizing them.

Understanding Polynomials: A Foundation for Factoring

Before we jump into the answer key, let's solidify our understanding of polynomials. A polynomial is an expression consisting of variables (usually represented by x, y, etc.) and coefficients, combined using addition, subtraction, and multiplication, but never division by a variable.

Key Polynomial Components:

• Terms: These are the individual parts of a polynomial separated by plus or minus signs. For example, in the polynomial 3x² + 2x - 5, the terms are 3x², 2x, and -5.
• Coefficients: These are the numerical values multiplying the variables. In 3x², the coefficient is 3.
• Variables: These are the letters representing unknown values (usually x).
• Exponents: These indicate the power to which the variable is raised. In 3x², the exponent is 2.
• Constant: A term without a variable (e.g., -5 in the example above).
• Degree: The highest exponent of the variable in the polynomial. The degree of 3x² + 2x - 5 is 2.

Factoring Polynomials: Breaking it Down

Factoring is the reverse of expanding (or multiplying) polynomials. It involves expressing a polynomial as a product of simpler polynomials. Mastering factoring is essential for solving equations, simplifying expressions, and understanding many advanced mathematical concepts.

Common Factoring Techniques:

• Greatest Common Factor (GCF): This is the first step in almost every factoring problem. Find the largest number and/or variable that divides all terms evenly.

  • Example: 6x³ + 9x² = 3x²(2x + 3) (The GCF is 3x²)

• Difference of Squares: This applies to binomials (two-term polynomials) of the form a² - b². It factors as (a + b)(a - b).

  • Example: x² - 16 = (x + 4)(x - 4)

• Perfect Square Trinomials: These are trinomials (three-term polynomials) that can be factored into the square of a binomial. They have the form a² + 2ab + b² or a² - 2ab + b².

  • Example: x² + 6x + 9 = (x + 3)²

• Factoring Trinomials (Quadratics): This involves finding two numbers that add up to the coefficient of the x term and multiply to the constant term.

  • Example: x² + 5x + 6 = (x + 2)(x + 3) (2 + 3 = 5 and 2 * 3 = 6)

• Grouping: Used for polynomials with four or more terms. Group terms with common factors and then factor out the GCF from each group.

  • Example: xy + 2x + 3y + 6 = x(y + 2) + 3(y + 2) = (x + 3)(y + 2)

Homework 7: A Step-by-Step Approach (Illustrative Examples)

Since I don't have access to your specific Homework 7, I will provide detailed solutions to a variety of problems that commonly appear in this unit. These examples will cover the factoring techniques mentioned above and will help you tackle your assignment effectively. Remember to always check your answers by expanding the factored form.

Problem 1: Factor 12x³ - 18x²

Solution: First, find the greatest common factor (GCF) of 12x³ and -18x². The GCF is 6x². Factor it out:

6x²(2x - 3)

Problem 2: Factor x² - 49

Solution: This is a difference of squares (a² - b², where a = x and b = 7). It factors as:

(x + 7)(x - 7)

Problem 3: Factor x² + 10x + 25

Solution: This is a perfect square trinomial. Notice that 25 is 5², and 10x is 2 * 5 * x. Therefore, it factors as:

(x + 5)²

Problem 4: Factor x² + 7x + 12

Solution: We need to find two numbers that add up to 7 (the coefficient of x) and multiply to 12 (the constant term). These numbers are 3 and 4. Therefore:

(x + 3)(x + 4)

Problem 5: Factor 2x³ + 4x² + x + 2

Solution: This requires factoring by grouping:

Group the first two terms and the last two terms: (2x³ + 4x²) + (x + 2)

Factor out the GCF from each group: 2x²(x + 2) + 1(x + 2)

Factor out the common binomial factor (x + 2): (2x² + 1)(x + 2)

Problem 6: Factor completely: 3x⁴ - 27x²

Solution: First, find the GCF. The GCF is 3x². Factor it out:

3x²(x² - 9)

Now, notice that (x² - 9) is a difference of squares:

3x²(x + 3)(x - 3)

Problem 7: Solve the equation x² - 5x + 6 = 0

Solution: First, factor the quadratic:

(x - 2)(x - 3) = 0

This equation is true if either (x - 2) = 0 or (x - 3) = 0. Therefore, the solutions are x = 2 and x = 3

Strategies for Mastering Polynomials and Factoring

• Practice Regularly: The key to mastering factoring is consistent practice. Work through numerous problems of varying difficulty.
• Understand the Concepts: Don't just memorize formulas; understand the underlying principles behind each factoring technique.
• Check Your Answers: Always expand your factored form to verify that it equals the original polynomial.
• Seek Help When Needed: Don't hesitate to ask your teacher, classmates, or tutor for help if you're struggling with a particular concept or problem.
• Use Online Resources: Utilize online resources such as Khan Academy, YouTube tutorials, and educational websites to supplement your learning. (But remember to avoid simply copying answers.)
• Break Down Complex Problems: If a problem seems overwhelming, break it down into smaller, more manageable steps.

By following these strategies and working through numerous practice problems, you'll significantly improve your understanding of polynomials and factoring. Remember, consistent effort and a deep understanding of the underlying concepts are the keys to success in algebra. Good luck with your Homework 7! Remember to always show your work, as the process is as important as the final answer. This demonstrates your understanding to your instructor and helps you identify any mistakes made during calculation.