Question 1 We can find the solutions of sin ⁡ 𝑥 = 0.3 sinx=0.3 algebraically. a) First we find the solutions in the interval [ 0 , 2 𝜋 ) [0,2π). We get one such solution by taking sin ⁡ − 1 sin −1 to get 𝑥 ≈ 0.30 x≈0.30. The other solution in this interval is 𝑥 ≈ 2.83 x≈2.83. b) We find all solutions by adding multiples of 2 𝜋 2π to the solutions in [ 0 , 2 𝜋 ) [0,2π). The solutions are 𝑥 ≈ 0.30 + 2 𝑘 𝜋 x≈0.30+2kπ and 𝑥 ≈ 2.83 + 2 𝑘 𝜋 x≈2.83+2kπ. Question 2 Grocery Shopping: Hot dogs are sold in packages of 10, but hot dog buns come in packs of eight. What is the smallest number that makes the hot dogs and buns come out even? Answer 1 Given information: Package of hot dogs contains 10. Package of hot dog buns contains 8. Concept used: The least common multiple (LCM) of two or more numbers is the smallest multiple that is completely divisible by the corresponding numbers, leaving no remainder. Calculations: Prime factorization of 10: 2 × 5 2×5 Prime factorization of 8: 2 × 2 × 2 2×2×2 The LCM is 2 × 2 × 2 × 5 = 40 2×2×2×5=40. Hence, the smallest number that makes hot dogs and buns come out even is 40. Question 3 A number that has exactly 2 factors (1 and itself) is called a ______ number. Options: factors prime composite multiplication Answer 1 The correct answer is: 2) prime A prime number has exactly two factors, 1 and itself. Question 4 Using Sum-to-Product Formulas, solve the equation by first using a Sum-to-Product Formula. sin ⁡ 𝜃 + sin ⁡ 3 𝜃 = 0 sinθ+sin3θ=0 Answer 1 Approach: Using the sum-to-product formula for sine: sin ⁡ 𝑢 + sin ⁡ 𝑣 = 2 sin ⁡ ( 𝑢 + 𝑣 2 ) cos ⁡ ( 𝑢 − 𝑣 2 ) sinu+sinv=2sin( 2 u+v )cos( 2 u−v ) Substitute 𝑢 = 𝜃 u=θ and 𝑣 = 3 𝜃 v=3θ: sin ⁡ 𝜃 + sin ⁡ 3 𝜃 = 2 sin ⁡ ( 4 𝜃 2 ) cos ⁡ ( − 2 𝜃 2 ) sinθ+sin3θ=2sin( 2 4θ )cos( 2 −2θ ) = 2 sin ⁡ ( 2 𝜃 ) cos ⁡ ( 𝜃 ) = 0 =2sin(2θ)cos(θ)=0 Using the zero product property: sin ⁡ 2 𝜃 = 0 sin2θ=0 cos ⁡ 𝜃 = 0 cosθ=0 For sin ⁡ 2 𝜃 = 0 sin2θ=0: 2 𝜃 = 𝑘 𝜋 ⇒ 𝜃 = 𝑘 𝜋 2 2θ=kπ⇒θ= 2 kπ For cos ⁡ 𝜃 = 0 cosθ=0: 𝜃 = 𝜋 2 + 𝑘 𝜋 θ= 2 π +kπ Thus, the solution is 𝜃 = 𝑘 𝜋 2 θ= 2 kπ for integer 𝑘 k. Question 5 A polynomial and one of its factors is given. Factor the polynomial COMPLETELY given that one of its factors is 𝑥 + 4 x+4. Your final answer should be in factored form. 𝑓 ( 𝑥 ) = 𝑥 7 + 4 𝑥 6 + 7 𝑥 4 + 28 𝑥 3 − 8 𝑥 − 32 f(x)=x 7 +4x 6 +7x 4 +28x 3 −8x−32 Answer 1 Step 1 Use synthetic division with 𝑥 + 4 x+4 as a factor: − 4 ∣ 1 4 0 7 0 28 − 8 − 32 −4∣1407028−8−32 After synthetic division, the quotient is 𝑥 6 + 7 𝑥 3 − 8 x 6 +7x 3 −8. Step 2 Factor 𝑥 6 + 7 𝑥 3 − 8 x 6 +7x 3 −8: 𝑥 6 + 7 𝑥 3 − 8 = ( 𝑥 3 + 8 ) ( 𝑥 3 − 1 ) x 6 +7x 3 −8=(x 3 +8)(x 3 −1) Using the sum and difference of cubes: ( 𝑥 + 2 ) ( 𝑥 2 − 2 𝑥 + 4 ) ( 𝑥 − 1 ) ( 𝑥 2 + 𝑥 + 1 ) (x+2)(x 2 −2x+4)(x−1)(x 2 +x+1) Final answer: ( 𝑥 + 4 ) ( 𝑥 + 2 ) ( 𝑥 2 − 2 𝑥 + 4 ) ( 𝑥 − 1 ) ( 𝑥 2 + 𝑥 + 1 ) (x+4)(x+2)(x 2 −2x+4)(x−1)(x 2 +x+1)