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

2021
Views:

5

Pages:

3
Author:

MathMonk

Questions and Answers Sheet 82 Polynomial Factorization Question #1 How do you multiply 3𝑥𝑦(−𝑥 4 − 4𝑥 2 𝑦 + 5𝑥𝑦 2 )? Answer: Explanation: 3𝑥𝑦(−𝑥 4 − 4𝑥 2 𝑦 + 5𝑥𝑦 2 ) = 3𝑥𝑦 ∗ (−𝑥 4 ) + 3𝑥𝑦 ∗ (−4𝑥 2 𝑦) + 3𝑥𝑦 ∗ (5𝑥𝑦 2 ) = −3𝑥 5 𝑦 − 12𝑥 3 ∗ 𝑦 2 + 15𝑥 2 𝑦 3 Question #2 𝑥 𝑥 5 3 How do you multiply ( ) ( ) ? Answer: Multiply the numerator of the first fraction times the numerator of the second fraction, and multiply the denominator of the first fraction times the denominator of the second fraction. Explanation: 𝑥 𝑥 ( )( ) = 5 3 𝑥∗𝑥 = 5∗3 2 𝑥 15 Question #3 How do you multiply (−3𝑡 2 + 4𝑡 − 7)(−2𝑡)? Answer: Explanation: The Distributive Property (of Multiplication over Addition) tells us: (𝑎 + 𝑏 + 𝑐) × (𝑑) = 𝑎𝑑 + 𝑏𝑑 + 𝑐𝑑 So (−3𝑡 2 + 4𝑡 − 7)(−2𝑡) = (−3𝑡 2 ) ∗ (−2𝑡) + (4𝑡) ∗ (−2𝑡) − (7) ∗ (−2𝑡) = 6𝑡 3 − 8𝑡 2 + 14𝑡 Answer: (−3𝑡 2 + 4𝑡 − 7)(−2𝑡) = 6𝑡 3 − 8𝑡 2 + 14𝑡 Question #4 How do you simplify 3𝑥 2 + 2𝑥(𝑥 − 5)? Answer: Explanation: 3𝑥 2 + 2𝑥(𝑥 − 5) Distribute the 2x by multiplying it times the terms inside the parentheses. 3𝑥 2 + 2𝑥 2 − 10𝑥 Add like terms. 5𝑥 2 − 10𝑥 Factor out the GCF 5x. 5𝑥(𝑥 − 2) Question #5 How do you simplify 4𝑎2 + 3𝑎 + (𝑎 + 7)2𝑎? Answer: Explanation: 4𝑎2 + 3𝑎 + (𝑎 + 7)2𝑎 = 4𝑎2 + 3𝑎 + 2𝑎2 + 14𝑎 Grouping like terms: = 4𝑎2 + 2𝑎2 + 3𝑎 + 14𝑎 = 6𝑎2 + 17𝑎 Question #6 How do you simplify 𝑥 3 + (5𝑥 2 − 6𝑥 − 1)𝑥 + 7𝑥 2 ? Answer: Explanation: 𝑥 3 + (5𝑥 2 − 6𝑥 − 1)𝑥 + 7𝑥 2 ⇒ 𝑥 3 + 5𝑥 3 − 6𝑥 2 − 𝑥 + 7𝑥 2 , removing bracket ⇒ 𝑥 3 + 5𝑥 3 + 7𝑥 2 − 6𝑥 2 − 𝑥 , bringing like terms together ⇒ 6𝑥 3 + 𝑥 2 − 𝑥 (Ans) Question #7 How do you multiply (𝑥 − 2)(3𝑥 2 − 5𝑥 + 4)? Answer: Explanation: You take the numbers in the first bracket, in this case, x and -2, and multiply both of those numbers with every single number in the other bracket. 3𝑥 3 − 5𝑥 2 + 4𝑥 − 6𝑥 2 + 10𝑥 − 8 Result: 3𝑥 3 − 11𝑥 2 + 14𝑥 − 8 Question #8 How do you simplify −2𝑐𝑑 2 (3𝑐 2 𝑑 − 2𝑐𝑑 2 + 𝑐𝑑 2 )? Answer: Explanation: −2𝑐𝑑 2 (3𝑐 2 𝑑 − 2𝑐𝑑 2 + 𝑐𝑑 2 ) ⇒ −2𝑐𝑑 2 (3𝑐 2 𝑑 − 𝑐𝑑 2 ) ⇒ (2𝑐𝑑 2 ∗ 𝑐𝑑)(3𝑐 − 2𝑑) ⇒ 2𝑐 2 𝑑 3 (3𝑐 − 2𝑑) = 6𝑐 3 𝑑 3 − 4𝑐 2 𝑑 4 Question #9 How do you simplify (𝑎3 + 2𝑎2 𝑏 − 𝑎𝑏 2 + 3𝑏 3 )(4𝑎𝑏)? Answer: Explanation: (𝑎3 + 2𝑎2 𝑏 − 𝑎𝑏 2 + 3𝑏 3 )(4𝑎𝑏) = (𝑎3 ) ∗ (4𝑎𝑏) + (2𝑎2 𝑏) ∗ (4𝑎𝑏) − (𝑎𝑏 2 ) ∗ (4𝑎𝑏) + (3𝑏 3 ) ∗ (4𝑎𝑏) = 4𝑎3 ∗ 𝑎𝑏 + 8𝑎2 𝑏 ∗ 𝑎𝑏 − 4𝑎𝑏 2 ∗ 𝑎𝑏 + 12𝑏 3 ∗ 𝑎𝑏 By property 𝑎𝑚 ∗ 𝑎𝑛 = 𝑎𝑚+𝑛 Applying the same to powers of a and b = 4𝑎(3+1) 𝑏 + 8𝑎(2+1) 𝑏 (1+1) − 4𝑎 (1+1) 𝑏 (2+1) + 12𝑎𝑏 3+1 = 4𝑎4 𝑏 + 8𝑎3 𝑏 2 − 4𝑎2 𝑏 3 + 12𝑎𝑏 4 = 4𝑎4 𝑏 + 4𝑎3 𝑏 2 + 12𝑎𝑏 4 Question #10 How do you find the product (𝑥 + 3)2 ? Answer: Explanation: (𝑥 + 3)2 = (𝑥 + 3) × (𝑥 + 3) = (𝑥 + 3) × (𝑥 + 3) = 𝑥 × (𝑥 + 3) + 3 × (𝑥 + 3) = 𝑥 2 + 3𝑥 + 3𝑥 + 9 = 𝑥 2 + 6𝑥 + 9