Questions and Answers #88 Polynomial Factorization

Questions and Answers Sheet 88 Polynomial Factorization Question #1 How do you find the product −2𝑑(𝑑 3 𝑐 2 − 4𝑑𝑐 2 + 2𝑑 2 𝑐) + 𝑐 2 (𝑑𝑐 2 − 3𝑑 4 )? Answer: First, expand both terms in parenthesis by multiplying each term within the parenthesis by the term outside the parenthesis: −2𝑑(𝑑 3 𝑐 2 − 4𝑑𝑐 2 + 2𝑑 2 𝑐) + 𝑐 2 (𝑑𝑐 2 − 3𝑑 4 ) ⇒ (−2𝑑 ∗ 𝑑 3 𝑐 2 ) − (−2𝑑 ∗ 4𝑑𝑐 2 ) + (−2𝑑 ∗ 2𝑑 2 𝑐) + (𝑐 2 ∗ 𝑑𝑐 2 ) − (𝑐 2 ∗ 3𝑑 4 ) ⇒ −2𝑑 4 𝑐 2 − (−8𝑑 2 𝑐 2 ) + (−4𝑑 3 𝑐) + 𝑑𝑐 4 − 3𝑑 4 𝑐 2 ⇒ −2𝑑 4 𝑐 2 + 8𝑑 2 𝑐 2 − 4𝑑 3 𝑐 + 𝑑𝑐 4 − 3𝑑 4 𝑐 2 Now, group and combine like terms: −2𝑑 4 𝑐 2 − 3𝑑 4 𝑐 2 + 8𝑑 2 𝑐 2 − 4𝑑 3 𝑐 + 𝑑𝑐 4 ⇒ (−2 − 3)𝑑 4 𝑐 2 + 8𝑑 2 𝑐 2 − 4𝑑 3 𝑐 + 𝑑𝑐 4 ⇒ −5𝑑 4 𝑐 2 + 8𝑑 2 𝑐 2 − 4𝑑 3 𝑐 + 𝑑𝑐 4 ⇒ −5𝑑 4 𝑐 2 − 4𝑑 3 𝑐 + 8𝑑 2 𝑐 2 + 𝑑𝑐 4 Question #2 How do you find the product −5𝑤 2 (8𝑤 2 𝑥 − 11𝑤𝑥 2 ) + 6𝑥(9𝑤𝑥 4 − 4𝑤 − 3𝑥 2 )? Answer: Explanation: −5𝑤 2 (8𝑤 2 𝑥 − 11𝑤𝑥 2 ) + 6𝑥(9𝑤𝑥 4 − 4𝑤 − 3𝑥 2 ) = −40𝑤 4 𝑥 + 55𝑤 3 𝑥 2 + 54𝑤𝑥 5 − 24𝑤𝑥 − 18𝑥 3 Question #3 How do you find the product 𝑏(𝑏 2 − 12𝑏 + 1)? Answer: Explanation: Multiply each term within the parenthesis by b: (𝑏 × 𝑏 2 ) − (𝑏 × 12𝑏) + (𝑏 × 1) Now we can use these two rules for exponents to finalize the solution: 𝑥1 = 𝑥 𝑥 𝑎 × 𝑥 𝑏 = 𝑥 𝑎+𝑏 (𝑏1 × 𝑏 2 ) − (𝑏1 × 12𝑏1 ) + (𝑏1 × 1) 𝑏1+2 − 12𝑏1+1 + 𝑏1 𝑏 3 − 12𝑏 2 + 𝑏 Question #4 How do you find the product 2𝑝𝑟 2 (2𝑝𝑟 + 5𝑝2 𝑟 − 15𝑝)? Answer: Explanation: 2𝑝𝑟 2 (2𝑝𝑟 + 5𝑝2 𝑟 − 15𝑝) ⇒ 2𝑝𝑟 2 ∗ 2𝑝𝑟 + 2𝑝𝑟 2 ∗ 5𝑝2 𝑟 − 2𝑝𝑟 2 ∗ 15𝑝 , removing the bracket ⇒ 4𝑝2 𝑟 3 + 10𝑝3 𝑟 3 − 30𝑝2 𝑟 2 Question #5 How do you multiply polynomials (𝑥 2 + 2𝑥 − 1)(𝑥 2 + 2𝑥 + 5)? Answer: Just use a modified version of foil or a table 𝑥 2 (𝑥 2 + 2𝑥 + 5) = 𝑥 4 + 2𝑥 3 + 5𝑥 2 2𝑥(𝑥 2 + 2𝑥 + 5) = 2𝑥 3 + 2𝑥 2 + 10𝑥 −1(𝑥 2 + 2𝑥 + 5) = −𝑥 2 − 2𝑥 − 5 Just add them all up 𝑥 4 + 2𝑥 3 + 5𝑥 2 + 2𝑥 3 + 2𝑥 2 + 10𝑥 − 𝑥 2 − 2𝑥 − 5 𝑥 4 + 2𝑥 3 + 2𝑥 3 + 5𝑥 2 + 2𝑥 2 − 𝑥 2 + 10𝑥 − 2𝑥 − 5 𝑥 4 + 4𝑥 3 + 6𝑥 2 + 8𝑥 − 5 Question #6 How do you multiply (𝑚2 − 4𝑚𝑝 + 𝑝2 )(𝑚2 + 4𝑚𝑝 − 𝑝2 )? Answer: Explanation: (𝑚2 − 4𝑚𝑝 + 𝑝2 )(𝑚2 + 4𝑚𝑝 − 𝑝2 ) = 𝑚2 (𝑚2 + 4𝑚𝑝 − 𝑝2 ) − 4𝑚𝑝(𝑚2 + 4𝑚𝑝 − 𝑝2 ) + 𝑝2 (𝑚2 + 4𝑚𝑝 − 𝑝2 ) = 𝑚4 + 4𝑚3 𝑝 − 𝑝2 𝑚2 − 4𝑚3 𝑝 − 16𝑚2 𝑝2 + 4𝑚𝑝3 + 𝑝2 𝑚2 + 4𝑚𝑝3 − 𝑝4 = 𝑚4 − 16𝑚2 𝑝2 + 8𝑚𝑝3 − 𝑝4 Question #7 How do you find the product −4𝑑(5𝑑 2 − 12) + 7(𝑑 + 5)? Answer: First, expand the terms in parenthesis by multiplying each term within the parenthesis by the term outside the parenthesis: −4𝑑(5𝑑 2 − 12) + 7(𝑑 + 5) (−4𝑑 ∗ 5𝑑 2 ) + (−4𝑑 ∗ −12) + (7 ∗ 𝑑) + (7 ∗ 5) −20𝑑 3 + 48𝑑 + 7𝑑 + 35 Now, combine like terms: −20𝑑 3 + (48 + 7)𝑑 + 35 −20𝑑 3 + 55𝑑 + 35 Question #8 How do you find the product 14𝑡 3 𝑢(2𝑡 2 𝑢2 − 10𝑡𝑢4 + 2)? Answer: Explanation: A product is the answer to a multiplication. We need to use the distributive law and multiply the factor by each term inside the bracket. 14𝑡 3 𝑢(2𝑡 2 𝑢2 − 10𝑡𝑢4 + 2) = 28𝑡 5 𝑢3 − 140𝑡 4 𝑢5 + 28𝑡 3 𝑢 Question #9 How do you find the product 𝑎(−8𝑎2 + 2𝑎 + 4) + 3(6𝑎2 − 4)? Answer: Explanation: First, expand both the terms in parenthesis by multiplying each term within the parenthesis by the term outside the parenthesis: 𝑎(−8𝑎2 + 2𝑎 + 4) + 3(6𝑎2 − 4) ⇒ (𝑎 × −8𝑎2 ) + (𝑎 × 2𝑎) + (𝑎 × 4) + (3 × 6𝑎2 ) − (3 × 4) ⇒ −8𝑎3 + 2𝑎2 + 4𝑎 + 18𝑎2 − 12 Next, group like terms: −8𝑎3 + 2𝑎2 + 18𝑎2 + 4𝑎 − 12 Now, combine like terms: −8𝑎3 + (2 + 18)𝑎2 + 4𝑎 − 12 −8𝑎3 + 20𝑎2 + 4𝑎 − 12 Question #10 How do you find the product −9𝑔(−2𝑔 + 𝑔2 ) + 3(𝑔2 + 4)? Answer: First, expand the terms in parenthesis by multiplying each term within the parenthesis by the term outside the parenthesis: −9𝑔(−2𝑔 + 𝑔2 ) + 3(𝑔2 + 4) ⇒ (−9𝑔 × −2𝑔) + (−9𝑔 × −𝑔2 ) + (3 × 𝑔2 ) + (3 × 4) ⇒ 18𝑔2 + (−9𝑔3 ) + 3𝑔2 + 12 ⇒ 18𝑔2 − 9𝑔3 + 3𝑔2 + 12 Next, group like terms: −9𝑔3 + 18𝑔2 + 3𝑔2 + 12 Now, combine like terms: −9𝑔3 + (18 + 3)𝑔2 + 12 −9𝑔3 + 21𝑔2 + 12