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

2023
Views:

18

Pages:

3
Author:

MathMonk

Questions and Answers Sheet 85 Polynomial Factorization Question #1 How do you simplify (𝑥 + 1)(𝑥 2 + 𝑥 + 1)? Answer: Explanation: To multiply these two terms you multiply each individual term in the left parenthesis by each individual term in the right parenthesis. (𝑥 + 1)(𝑥 2 + 𝑥 + 1) becomes: (𝑥 × 𝑥 2 ) + (𝑥 × 𝑥) + (𝑥 × 1) + (1 × 𝑥 2 ) + (1 × 𝑥) + (1 × 1) 𝑥3 + 𝑥2 + 𝑥 + 𝑥2 + 𝑥 + 1 We can now group and combine like terms: 𝑥3 + 𝑥2 + 𝑥2 + 𝑥 + 𝑥 + 1 𝑥 3 + 1𝑥 2 + 1𝑥 2 + 1𝑥 + 1𝑥 + 1 𝑥 3 + (1 + 1)𝑥 2 + (1 + 1)𝑥 + 1 𝑥 3 + 2𝑥 2 + 2𝑥 + 1 Question #2 How do you simplify 3𝑥 2 (3𝑥 2 − 𝑥 − 2)? Answer: Explanation: To simplify this expression you need to multiply the term outside the parenthesis (3𝑥 2 ) by each term within the parenthesis: (3𝑥 2 ∗ 3𝑥 2 ) − (3𝑥 2 ∗ 𝑥) − (3𝑥 2 ∗ 2) (3 ∗ 3 ∗ 𝑥 2 ∗ 𝑥 2 ) − (3 ∗ 𝑥 2 ∗ 𝑥) − (3 ∗ 2 ∗ 𝑥 2 ) We can now use the following rule for exponents to multiply the individual terms: 𝑥 𝑎 ∗ 𝑥 𝑏 = 𝑥 𝑎+𝑏 Hint: remember 𝑥 = 𝑥 1 9𝑥 2+2 − 3𝑥 2+1 − 6𝑥 2 9𝑥 4 − 3𝑥 3 − 6𝑥 2 Question #3 How do you simplify −3𝑥 2 (𝑥 2 + 3𝑥) ? Answer: To simplify this expression multiply each term within the parenthesis by the term outside the parenthesis: −3𝑥 2 (𝑥 2 + 3𝑥) ⇒ (−3𝑥 2 ∗ 𝑥 2 ) + (−3𝑥 2 ∗ 3𝑥) ⇒ −3𝑥 4 − 9𝑥 3 Question #4 How do you multiply (4𝑥 − √3)(4𝑥 + √3)? Answer: Explanation: (4𝑥 − √3)(4𝑥 + √3) Use FOIL - firsts, outers, inners, lasts first: 4𝑥 ∗ 4𝑥 = 16𝑥 2 outer: 4𝑥 ∗ √3 = 4√3𝑥 inner: −√3 ∗ 4𝑥 = −4√3𝑥 lasts: −√3 ∗ √3 = −3 Then add all the results 16𝑥 2 + 4√3 − 4√3 − 3 The 4√3 and −4√3 cancel each other out so you are left with 16𝑥 2 − 3 Question #5 How do you find the product of −2𝑥(𝑥 2 − 3)? Answer: Explanation: Multiply. 𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐 −2𝑥(𝑥 2 − 3) −2𝑥(𝑥 2 ) + 2𝑥(3) −2𝑥 3 + 6𝑥 Question #6 How do you multiply 7𝑐(5 − 𝑦 + 4𝑦 2 )? Answer: To multiply this expression you need to multiply every term in the parenthesis by 7c 7𝑐(5 − 𝑦 + 4𝑦 2 ) becomes: (7𝑐 × 5) − (7𝑐 × 𝑦) + (7𝑐 × 4𝑦 2 ) → 35𝑐 − 7𝑐𝑦 + 28𝑐𝑦 2 Or, writing it in standard form: 28𝑐𝑦 2 − 7𝑐𝑦 + 35𝑐 Question #7 How do you multiply −4𝑏(1 − 9𝑏 − 2𝑏 2 )? Answer: Explanation: Just Distribute -4b and multiply with each term. −4𝑏(1 − 9𝑏 − 2𝑏 2 ) = (−4𝑏)(1) − (−4𝑏)(9𝑏) − (−4𝑏)(2𝑏 2 ) Watch out for Sign Changes, if any. = −4𝑏 2 + 36𝑏 2 + 8𝑏 3 Re-order from highest power to lowest: = 8𝑏 3 + 36𝑏 2 − 4𝑏 Question #8 How do you multiply 2𝑚2 (2𝑚2 + 3𝑚 − 5)? Answer: Explanation: Distribute the term outside of the set of parenthesis to each term on the inside of the parenthesis. So, start with the first term in the parenthesis: 2𝑚2 Distribute 2𝑚2 to 2𝑚2 and multiply 2𝑚2 × 2𝑚2 = 4𝑚4 Next distribute 2𝑚2 to the second term in the parenthesis: 3m 2𝑚2 × 3𝑚 = 6𝑚3 Lastly, distribute 2𝑚2 to the third and final term in the parenthesis: -5 2𝑚2 × −5 = −10𝑚2 So, the equation would look like: 4𝑚4 + 6𝑚3 − 102 Question #9 How do you multiply 3(𝑥 2 + 2𝑥 + 6)? Answer: Multiply each term by 3 Explanation: take 𝑥 2 ∗ 3 and 2𝑥 ∗ 3 and 6 ∗ 3 so the answer would be 3𝑥 2 + 6𝑥 + 18 Question #10 How do you expand (4𝑥 − 3)2 ? Answer: Explanation: Given: (4𝑥 − 3)2 Use the binomial theorem for the second power, which we find that: (𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏 2 (𝑎 − 𝑏)2 = 𝑎2 − 2𝑎𝑏 + 𝑏 2 Here 𝑎 = 4𝑥, 𝑏 = 3, and we use the second formula. So, we get: (4𝑥 − 3)2 = (4𝑥)2 − 2 ∗ 3 ∗ 4𝑥 + 32 = 16𝑥 2 − 24𝑥 + 9