University:
Rice University
Course:
MATH 463 | Advanced Algebra I
Academic year:

2023
Views:

28

Pages:

3
Author:

MathMonk

Questions and Answers Sheet 96 Polynomial Factorization Question #1 How do you find the product of (2ℎ + 3)(2ℎ2 + 3ℎ + 4)? Answer: Each term in the 2nd bracket must be multiplied by each term in the 1st bracket. The following shows how this may be done. (2ℎ + 3)(2ℎ2 + 3ℎ + 4) = 2ℎ(2ℎ2 + 3ℎ + 4) + 3(2ℎ2 + 3ℎ + 4) now distribute each pair of brackets. = 4ℎ3 + 6ℎ2 + 8ℎ + 6ℎ2 + 9ℎ + 12 and collecting like terms gives. = 4ℎ3 + (6ℎ2 + 6ℎ2 ) + (8ℎ + 9ℎ) + 12 = 4ℎ3 + 12ℎ2 + 17ℎ + 12 ← result of product Question #2 How do you simplify (2𝑦 + 5𝑥)2 ? Answer: Combine the like terms in the "middle" : 4𝑦 2 + 20𝑥𝑦 + 25𝑥 2 If you want a challenge, you can do this in your head: 1) square the first term 2y: 22 𝑦 2 = 4𝑦 2 2) Square the first terms: 2𝑦 ∗ 5𝑥 = 10𝑥𝑦 and double it 2 ∗ 10𝑥𝑦 = 20𝑥𝑦 3) Square the second term: 52 𝑥 2 = 25𝑥 2 4) and combine. 4𝑦 2 + 20𝑥𝑦 + 25𝑥 2 Question #3 How do you multiply (𝑥 + 4)(𝑥 2 + 2𝑥 − 5)? Answer: (𝑥 + 4)(𝑥 2 + 2𝑥 − 5) Here, we first multiply x contained in the first bracket with all three terms of second bracket: = 𝑥 ∗ (𝑥 2 ) + 𝑥 ∗ (2𝑥) + 𝑥 ∗ (−5) = 𝑥 3 + 2𝑥 2 − 5𝑥 … (1) Now, we multiply 4 contained in the first bracket with all three terms of second bracket: = 4 ∗ (𝑥 2 ) + 4 ∗ 2𝑥 + 4 ∗ (−5) = 4𝑥 2 + 8𝑥 − 20 … (2) combining (1) and (2) = 𝑥 3 + 6𝑥 2 + 3𝑥 − 20 Question #4 How do you multiply (3𝑥 + 2)(𝑥 2 + 7𝑥 − 6)? Answer: Explanation: Just use distribution: (𝑎 + 𝑏) ∗ (𝑐) + 𝑏 ∗ (𝑐) 3𝑥(𝑥 2 + 7𝑥 − 6) + 2(𝑥 2 + 7𝑥 − 6) = 3𝑥 3 + 21𝑥 2 − 18𝑥 + 2𝑥 2 + 14𝑥 − 12 Question #5 How do you multiply (2𝑥 2 − 5𝑥 + 10)(𝑥 + 4)? Answer: Explanation: (2𝑥 2 − 5𝑥 + 10)(𝑥 + 4) Each of the three terms within the first bracket need to be multiplied with both terms of the second bracket. = (2𝑥 2 − 5𝑥 + 10)(𝑥) + (2𝑥 2 − 5𝑥 + 10)(4) = 2𝑥 3 − 5𝑥 2 + 10𝑥 + 8𝑥 2 − 20𝑥 + 40 grouping like terms = 2𝑥 3 + (8𝑥 2 − 5𝑥 2 ) + (10𝑥 − 20𝑥) + 40 = 2𝑥 3 + 3𝑥 2 − 10𝑥 + 40 Question #6 How do you simplify (7𝑦 − 𝑥)2 ? Answer: Explanation: It is a binomial squared: (𝑎 + 𝑏)2 = 𝑎2 + 𝑏 2 + 2𝑎𝑏 Answer: 49𝑦 2 + 𝑥 2 − 14𝑥𝑦 Question #7 How do you multiply (3𝑡 2 − 2𝑤)2 ? Answer: Explanation: Use the quadratic expansion: (𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏 2 with 𝑎 = 3𝑡 2 and 𝑏 = −2𝑤 This gives: (3𝑡 2 − 2𝑤)2 = 9𝑡 4 − 12𝑡 2 𝑤 + 4𝑤 2 Question #8 How do you multiply (5𝑛 − 4)2 ? Answer: Explanation: (𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏 2 2 (5𝑛 + (−4)) + (5𝑛)2 + 2 ∗ 5𝑛 ∗ (−4) + (−4)2 Question #9 How do you multiply 3𝑦 2 𝑧(2𝑦 2 𝑧 + 4𝑦𝑧 − 𝑦 + 𝑧)? Answer: 3𝑦 2 𝑧(2𝑦 2 𝑧 + 4𝑦𝑧 − 𝑦 + 𝑧) = 3𝑦 2 𝑧(2𝑦 2 𝑧) + 3𝑦 2 𝑧 ∗ (4𝑦𝑧) − 3𝑦 2 𝑧 ∗ (𝑦) + 3𝑦 2 𝑧 ∗ (𝑧) = 6𝑦 4 𝑧 2 + 12𝑦 3 𝑧 2 − 3𝑦 3 𝑧 + 3𝑦 2 𝑧 2 Question #10 How do you multiply (−13𝑢 + 𝑣 2 )2 ? Answer: The formula for (𝑎 + 𝑏)2 is 𝑎2 + 2𝑎𝑏 + 𝑏 2 . Let's apply this to our problem. Explanation: (−13𝑢 + 𝑣 2 )2 = (−13𝑢)2 + 2(−13𝑢)(𝑣 2 ) + (𝑣 2 )2 = 13𝑢 ∗ −13𝑢 = 169𝑢2 2(−13𝑢)(𝑣 2 ) = −26𝑢𝑣 2 𝑣2 ∗ 𝑣2 = 𝑣4 Therefore, our answer is 169𝑢2 − 26𝑢𝑣 2 + 𝑣 4 .