Polynomial Analysis

lessonbunny.com

Answer Key & Explanations

Polynomial Analysis · Grade 12 · Worksheet 1

1. lim(x→∞) (4x⁵ - 3x³ + 7x - 2)/(2x⁵ - 5x² + 9) = ? Answer: 2 Solution: Identify the degrees of numerator and denominator. Both have degree 5. Identify the leading coefficients.
   Full step-by-step solution

   Step 1: Identify the degrees of numerator and denominator. Both have degree 5. Step 2: Identify the leading coefficients. Numerator leading coefficient is 4, denominator leading coefficient is 2. Step 3: For rational functions where degrees are equal, the limit at infinity equals the ratio of leading coefficients: 4/2 = 2. Step 4: Therefore, lim(x→∞) (4x⁵ - 3x³ + 7x - 2)/(2x⁵ - 5x² + 9) = 2.

2. Mere is analyzing the long-term behavior of a polynomial function that models the height in meters of a drone over time in hours. The function is f(x) = -6x^8 + 4x^3 - 2. Describe the end behavior of this polynomial as x approaches positive and negative infinity. Answer: As x → -∞, f(x) → -∞; as x → +∞, f(x) → -∞ Solution: Identify the leading term. The polynomial is f(x) = -6x^8 + 4x^3 - 2. The leading term is -6x^8, because it has the highest exponent.
   Full step-by-step solution

   Step 1: Identify the leading term. The polynomial is f(x) = -6x^8 + 4x^3 - 2. The leading term is -6x^8, because it has the highest exponent. Step 2: Determine the degree. The degree is 8, which is an even number. Step 3: Determine the leading coefficient. The leading coefficient is -6, which is negative. Step 4: Apply end behavior rules. For an even-degree polynomial: if the leading coefficient is positive, both ends go up; if negative, both ends go down. Since the degree is even and the coefficient is negative, as x → -∞, f(x) → -∞, and as x → +∞, f(x) → -∞. Final answer: As x → -∞, f(x) → -∞; as x → +∞, f(x) → -∞.

3. f(x) = -6x¹¹ + 21x⁶ - 4x³ + 1. Describe the end behavior as x → ∞ and x → -∞. Answer: As x → ∞, f(x) → -∞; as x → -∞, f(x) → ∞ Solution: Identify the leading term. The highest power is x¹¹, so the leading term is -6x¹¹. Degree = 11 (odd), leading coefficient = -6 (negative).
   Full step-by-step solution

   Step 1: Identify the leading term. The highest power is x¹¹, so the leading term is -6x¹¹. Degree = 11 (odd), leading coefficient = -6 (negative). Step 2: For odd degree with negative leading coefficient: as x → ∞, the term -6x¹¹ dominates and is negative large → f(x) → -∞. As x → -∞, x¹¹ is negative (since odd power), so -6 × (negative) = positive large → f(x) → ∞. Step 3: Therefore, end behavior: as x → ∞, f(x) → -∞; as x → -∞, f(x) → ∞.

4. Olivia is modeling the long-term behavior of a company's profit (in millions of dollars) using the polynomial function P(x) = -7x^9 + 3x^5 - 2x^2 + 11. As the number of years x becomes very large (x → ∞) or very negative (x → -∞), what can she conclude about the company's profit? Describe the end behavior of P(x) as x → ∞ and as x → -∞. Answer: As x → ∞, P(x) → -∞; as x → -∞, P(x) → ∞ Solution: Identify the leading term of P(x) = -7x^9 + 3x^5 - 2x^2 + 11. The leading term is -7x^9, because it has the highest exponent (degree 9). Step 2: Determine the degree: 9 is odd.
   Full step-by-step solution

   Step 1: Identify the leading term of P(x) = -7x^9 + 3x^5 - 2x^2 + 11. The leading term is -7x^9, because it has the highest exponent (degree 9). Step 2: Determine the degree: 9 is odd. Step 3: Determine the leading coefficient: -7 is negative. Step 4: For a polynomial with odd degree and negative leading coefficient: As x → ∞ (large positive), the leading term -7x^9 becomes very large negative, so P(x) → -∞. As x → -∞ (large negative), x^9 is negative (since odd power), and -7 times a negative is positive, so P(x) → ∞. Final answer: As x → ∞, P(x) → -∞; as x → -∞, P(x) → ∞.

5. Emma is analyzing the long-term behavior of a polynomial function that models the profit (in millions of dollars) of a company over time. The function is P(x) = -5x^7 + 4x^5 - 2x^3 + x - 10, where x represents years since 2020. As x → ∞ (far into the future) and as x → -∞ (looking back in time), what happens to the profit? Describe the end behavior of P(x) using the degree and leading coefficient. Answer: As x → ∞, P(x) → -∞; as x → -∞, P(x) → ∞ Solution: Identify the leading term. The polynomial is P(x) = -5x^7 + 4x^5 - 2x^3 + x - 10. The highest power is x^7, so the leading term is -5x^7.
   Full step-by-step solution

   Step 1: Identify the leading term. The polynomial is P(x) = -5x^7 + 4x^5 - 2x^3 + x - 10. The highest power is x^7, so the leading term is -5x^7. Step 2: Determine the degree. The degree is 7, which is odd. Step 3: Determine the sign of the leading coefficient. The leading coefficient is -5, which is negative. Step 4: For an odd-degree polynomial with a negative leading coefficient: as x → ∞ (positive direction), the function goes to -∞; as x → -∞ (negative direction), the function goes to +∞. Step 5: Write the end behavior: As x → ∞, P(x) → -∞. As x → -∞, P(x) → ∞. The answer is: As x → ∞, P(x) → -∞; as x → -∞, P(x) → ∞.

6. Ava is studying the growth pattern of a particular species of bacteria for her biology project. She models the population (in thousands) over time (in hours) using the polynomial function P(t) = -2t^6 + 5t^3 - 7t + 11. As time goes on, Ava wants to understand what happens to the bacterial population in the very long term. Describe the end behavior of this polynomial function as t approaches positive infinity and as t approaches negative infinity. Answer: As t → +∞, P(t) → -∞; as t → -∞, P(t) → -∞ Solution: Identify the leading term of the polynomial P(t) = -2t^6 + 5t^3 - 7t + 11. The leading term is -2t^6. Determine the degree.
   Full step-by-step solution

   Step 1: Identify the leading term of the polynomial P(t) = -2t^6 + 5t^3 - 7t + 11. The leading term is -2t^6. Step 2: Determine the degree. The exponent of the leading term is 6, which is even. Step 3: Determine the leading coefficient. The coefficient is -2, which is negative. Step 4: For an even-degree polynomial with a negative leading coefficient, the end behavior is: as x → +∞, y → -∞ and as x → -∞, y → -∞. Step 5: Apply this to the variable t: as t → +∞, P(t) → -∞; as t → -∞, P(t) → -∞. Final answer: As t approaches positive infinity, P(t) approaches negative infinity. As t approaches negative infinity, P(t) also approaches negative infinity.

7. f(x) = -8x⁷ + 13x⁵ - 9x³ + 4x - 11. Describe the end behavior as x → ∞ and x → -∞. Answer: As x → ∞, f(x) → -∞; as x → -∞, f(x) → ∞ Solution: Identify the leading term. The leading term is -8x⁷ because it has the highest exponent (degree 7). Determine the degree.
   Full step-by-step solution

   Step 1: Identify the leading term. The leading term is -8x⁷ because it has the highest exponent (degree 7). Step 2: Determine the degree. The degree is 7, which is odd. Step 3: Determine the leading coefficient. The leading coefficient is -8, which is negative. Step 4: For an odd-degree polynomial with a negative leading coefficient: - As x → ∞ (right end), f(x) → -∞ (the graph falls to the right). - As x → -∞ (left end), f(x) → ∞ (the graph rises to the left). The answer is: As x → ∞, f(x) → -∞; as x → -∞, f(x) → ∞.