unit 7 polynomials and factoring homework 1 answer key

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01-02-2025

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Unit 7 Polynomials and Factoring: Homework 1 - Answer Key & Concepts Review

This guide provides answers and a comprehensive review of the key concepts covered in Homework 1 for Unit 7 on Polynomials and Factoring. Remember, understanding the why behind the answers is crucial for mastering this topic. We'll break down each problem type, offering explanations to solidify your understanding.

Note: Since I do not have access to your specific homework assignment, I will provide a general answer key covering common polynomial and factoring problems encountered in this unit. Adapt these examples to your specific problems. If you have specific questions from your assignment, please provide them and I will gladly help.

Section 1: Polynomial Addition and Subtraction

Key Concept: Combining like terms. Remember, you can only add or subtract terms with the same variable and exponent.

Example Problem: Simplify (3x² + 2x - 5) + (x² - 4x + 7)

Answer: Combine like terms: (3x² + x²) + (2x - 4x) + (-5 + 7) = 4x² - 2x + 2

Example Problem: Subtract (5x³ - 2x + 1) from (8x³ + 3x² - x + 6)

Answer: This is equivalent to (8x³ + 3x² - x + 6) - (5x³ - 2x + 1). Distribute the negative sign: 8x³ + 3x² - x + 6 - 5x³ + 2x - 1. Combine like terms: (8x³ - 5x³) + 3x² + (-x + 2x) + (6 - 1) = 3x³ + 3x² + x + 5

Section 2: Polynomial Multiplication

Key Concept: Distributive property (FOIL method for binomials).

Example Problem: Multiply (2x + 3)(x - 4)

Answer: Using the FOIL method (First, Outer, Inner, Last): (2x)(x) + (2x)(-4) + (3)(x) + (3)(-4) = 2x² - 8x + 3x - 12 = 2x² - 5x - 12

Example Problem: Multiply (x + 2)(x² - 3x + 1)

Answer: Distribute each term in the first binomial to each term in the second binomial: x(x² - 3x + 1) + 2(x² - 3x + 1) = x³ - 3x² + x + 2x² - 6x + 2. Combine like terms: x³ - x² - 5x + 2

Section 3: Factoring Polynomials

Key Concepts: Greatest Common Factor (GCF), factoring quadratics, difference of squares.

Example Problem (GCF): Factor 6x² + 12x

Answer: The GCF of 6x² and 12x is 6x. Factor it out: 6x(x + 2)

Example Problem (Factoring Quadratics): Factor x² + 5x + 6

Answer: Find two numbers that add up to 5 (the coefficient of x) and multiply to 6 (the constant term). These numbers are 2 and 3. Therefore, the factored form is (x + 2)(x + 3)

Example Problem (Difference of Squares): Factor x² - 25

Answer: This is a difference of squares (a² - b² = (a + b)(a - b)). Here, a = x and b = 5. The factored form is (x + 5)(x - 5)

Section 4: Solving Polynomial Equations

Key Concept: Setting the equation equal to zero and factoring to find the roots (solutions).

Example Problem: Solve x² - 7x + 12 = 0

Answer: Factor the quadratic: (x - 3)(x - 4) = 0. Set each factor equal to zero: x - 3 = 0 or x - 4 = 0. Solve for x: x = 3 or x = 4

This comprehensive review covers the fundamental concepts of Unit 7, Polynomials and Factoring. Remember to practice regularly and consult your textbook or teacher if you have further questions. Good luck!