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

2021
Views:

29

Pages:

3
Author:

MathMonk

Questions and Answers Sheet 61 Polynomial Factorization Question #1 3 How do you write 𝑦 = − 𝑥 + 3 in standard form? 2 Answer: Explanation: the equation of a line in standard form is. Ax+By=C where A is a positive integer and B, C are integers 3 given 𝑦 = − 𝑥 + 3 2 multiply all terms by 2 2y=-3x+6 add 3x to both sides 3𝑥 + 2𝑦 = 6 ← in standard form Question #2 How do you write 𝑥 2 − 8𝑦 2 − 14𝑥 − 15 = 0 in standard form? Answer: This is a hyperbola with a horizontal transverse axis so its standard form is (𝑥−ℎ)2 𝑎2 − (𝑦−𝑘)2 2 𝑏2 =1 Given 𝑥 − 8𝑦 2 − 14𝑥 − 15 = 0 𝑥 2 − 14𝑥 + 49 − 8𝑦 2 = 15 + 49 (𝑥 − 7)2 − 8𝑦 2 = 64 (𝑥−7)2 82 − 𝑦2 2 (2√2) =1 Question #3 How do you write 6𝑥 2 − 24𝑥 − 5𝑦 2 − 10𝑦 − 11 = 0 in standard form? Answer: The standard form of a hyperbola with a transverse axis (all stuff this is, trust me on this) is (𝑥−ℎ)2 𝑎2 2 − (𝑦−𝑘)2 𝑏2 =1 6𝑥 − 24𝑥 − 5𝑦 2 − 10𝑦 − 11 = 0 move he constant to the left side 6(𝑥 2 − 4𝑥) − 5(𝑦 2 + 2𝑦) = 11 complete the squares 6(𝑥 2 − 4𝑥 + 4) − 5(𝑦 2 + 2𝑦 + 1) = 11 + 24 − 5 6(𝑥 − 2)2 − 5(𝑦 + 1)2 = 30 Divide by 30 so left side equals 1 (𝑥−2)2 (√5) 2 − (𝑦+1)2 2 (√6) =1 Question #4 How do you write 25𝑦 2 + 9𝑥 2 − 50𝑦 − 54𝑥 = 119 in standard form? Answer: 25𝑦 2 + 9𝑥 2 − 50𝑦 − 54𝑥 = 119 Consider the tow sub-expressions from the left side of this equation: 1. Terms involving y 25𝑦 2 − 50𝑦 = 25(𝑦 2 − 2𝑦) = 25(𝑦 2 − 2𝑦 + 1) − 25 = 52 (𝑦 − 1)2 − 25 2. Terms involving x 9𝑥 2 − 54𝑥 = 9(𝑥 2 − 6𝑥 + (−3)2 ) − 81 = 32 (𝑥 − 3)2 − 81 25𝑦 2 + 9𝑥 2 − 50𝑦 − 54𝑥 = 119 25𝑦 2 − 50𝑦 + 9𝑥 2 − 54𝑥 = 119 = 52 (𝑦 − 1)2 − 25 + 32 (𝑥 − 3)2 − 81 = 119 = 52 (𝑦 − 1)2 + 32 (𝑥 − 3)2 = 225 or = (25𝑦 − 25)2 + (9𝑥 − 27)2 = 152 or some variant of this depending upon your local definition of "standard form" for an ellipse. Question #5 How do you write 3𝑥 = 4 − 3𝑥 2 in standard form? Answer: Standard form puts the terms in descending order of exponent. Add (3𝑥 2 − 4) to both sides to get: 3𝑥 2 + 3𝑥 − 4 = 0 Question #6 1 1 3 4 How do you write 𝑦 + = (𝑥 − 3) in standard form? Answer: 1 1 3 4 𝑦 + = (𝑥 − 3) To get rid of the fractions, multiply both sides 12: 1 1 3 4 12 (𝑦 + ) = 12 × (𝑥 − 3) 12y+4=3(x-3) 12y+4=3x-9 Standard form : Ax+By=C Rearrange to get: 3x-12y=13 Question #7 What is the standard form for 𝑦 = 6(𝑥 − 2)2 − 8 ? Answer: Standard form of 𝑦 = 6(𝑥 − 2)2 − 8 Explanation: To find the standard form, develop the vertex form. 𝑦 = 6(𝑥 2 − 4𝑥 + 4) − 8 = 6𝑥 2 − 24𝑥 + 16 Question #8 What is the standard form for 𝑦 = 4(𝑥 − 2)2 − 1 ? Answer: Explanation: 𝑦 = 4(𝑥 − 2)2 − 1 𝑦 = 4(𝑥 2 − 4𝑥 + 4) − 1 expand binomial product 𝑦 = 4𝑥 2 − 14𝑥 + 16 − 1 expand brackets 𝑦 = 4𝑥 2 − 14𝑥 + 15 simplify Answer: 𝑦 = 4𝑥 2 − 14𝑥 + 15 Question #9 How do you write 𝑦 = 2(𝑥 + 4)2 − 21 in standard form? Answer: Standard form relates to writing a function such as yours as 𝑎𝑥 𝑛 + 𝑏𝑥 𝑛−1 + ⋯ + 𝑎′ 𝑦 𝑛 + 𝑏 ′ 𝑦 𝑛−1 + ⋯ constant Thus, let's just distribute your product and then move everything to the same side: 𝑦 = 2(𝑥 2 + 8𝑥 + 16) − 21 𝑦 = 2𝑥 2 + 16𝑥 + 32 − 21 𝑦 = 2𝑥 2 + 16𝑥 + 11 2𝑥 2 + 16𝑥 − 𝑦 + 11 = 0 Question #10 4 6 5 5 How do you write 𝑦 = − ( ) 𝑥 + ( ) in standard form? Answer: Standard form of a line: Ax+By=C 4 6 5 5 𝑦 = −( )𝑥 + ( ) Multiply both sides by 5: 5y=-4x+6 Rearrange: 4x+5y=6