Polynomial Analysis

lessonbunny.com

Answer Key & Explanations

Polynomial Analysis · Grade 12 · Worksheet 2

1. f(x) = -5x⁷ + 20x³ - 15. Describe the end behavior as x → ∞ and x → -∞. Answer: As x → ∞, f(x) → -∞; as x → -∞, f(x) → ∞ Solution: Identify the leading term. The polynomial is f(x) = -5x⁷ + 20x³ - 15. The leading term is -5x⁷.
   Full step-by-step solution

   Step 1: Identify the leading term. The polynomial is f(x) = -5x⁷ + 20x³ - 15. The leading term is -5x⁷. Step 2: Determine the degree. The degree is 7, which is odd. Step 3: Determine 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 → ∞, f(x) → -∞, and as x → -∞, f(x) → ∞. Step 5: Therefore, the end behavior is: as x → ∞, f(x) → -∞; as x → -∞, f(x) → ∞.

2. Kaia is sketching the graph of the polynomial function f(x) = -9x^8 + 14x^5 - 22x^2 + 7 on a coordinate plane. Describe the end behavior of this polynomial by completing the statements: As x → -∞, f(x) → ? and As x → +∞, f(x) → ? Answer: As x → -∞, f(x) → -∞ and As x → +∞, f(x) → -∞ Solution: Identify the leading term. The leading term is the term with the highest power of x, which is -9x^8. Determine the degree.
   Full step-by-step solution

   Step 1: Identify the leading term. The leading term is the term with the highest power of x, which is -9x^8. Step 2: Determine the degree. The exponent of the leading term is 8, which is even. Step 3: Determine the leading coefficient. The coefficient of the leading term is -9, which is negative. Step 4: Recall the end behavior rules for polynomials: For an even degree, both ends go in the same direction. If the leading coefficient is positive, both ends go up to positive infinity. If the leading coefficient is negative, both ends go down to negative infinity. Step 5: Apply these rules: even degree (8) and negative leading coefficient (-9) mean both ends go down to negative infinity. Thus, as x → -∞, f(x) → -∞ and as x → +∞, f(x) → -∞.

3. lim(x→∞) (4x⁵ - 3x³ + 7x - 2)/(2x⁵ - x² + 5) = ? Answer: 2 Solution: Identify the highest degree terms in numerator and denominator Numerator highest degree: 4x⁵ Denominator highest degree: 2x⁵ For limits at infinity of rational functions, only the highest degree terms matter lim(x→∞) (4x⁵ - 3x³ + 7x - 2)/(2x⁵ - x² + 5) = lim(x→∞) (4x⁵)/(2x⁵) (4x⁵)/(2x⁵) = 4/2 =…
   Full step-by-step solution

   Step 1: Identify the highest degree terms in numerator and denominator Numerator highest degree: 4x⁵ Denominator highest degree: 2x⁵ Step 2: For limits at infinity of rational functions, only the highest degree terms matter lim(x→∞) (4x⁵ - 3x³ + 7x - 2)/(2x⁵ - x² + 5) = lim(x→∞) (4x⁵)/(2x⁵) Step 3: Simplify the ratio of highest degree terms (4x⁵)/(2x⁵) = 4/2 = 2 Step 4: Therefore, the limit equals 2 lim(x→∞) (4x⁵ - 3x³ + 7x - 2)/(2x⁵ - x² + 5) = 2

4. Noah is analyzing the long-term behavior of a polynomial function that models the profit (in thousands of dollars) of a company over time, where x represents years since 2000. The function is P(x) = -7x^9 + 12x^5 - 3x^2 + 8. Describe the end behavior of this polynomial function as x approaches positive infinity and as x approaches negative infinity. Answer: As x → -∞, P(x) → +∞; as x → +∞, P(x) → -∞ Solution: Identify the leading term. The polynomial is P(x) = -7x^9 + 12x^5 - 3x^2 + 8. The leading term is the term with the highest exponent, which is -7x^9.
   Full step-by-step solution

   Step 1: Identify the leading term. The polynomial is P(x) = -7x^9 + 12x^5 - 3x^2 + 8. The leading term is the term with the highest exponent, which is -7x^9. The degree is 9 (odd), and the leading coefficient is -7 (negative). Step 2: For an odd-degree polynomial with a negative leading coefficient, as x → +∞, the function approaches -∞. This is because a positive x raised to an odd power is positive, but multiplied by a negative coefficient makes it negative, and as x gets larger, the magnitude grows without bound in the negative direction. Step 3: As x → -∞, a negative x raised to an odd power is negative, and multiplying by a negative coefficient (-7) gives a positive result. Thus, as x → -∞, the function approaches +∞. Step 4: Therefore, the end behavior is: as x → -∞, P(x) → +∞; as x → +∞, P(x) → -∞.

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

   Step 1: Identify the degrees of numerator and denominator. Both are degree 5. Step 2: Identify the leading coefficients. Numerator leading coefficient is 4, denominator leading coefficient is 2. Step 3: When 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 - 1) = 2.

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) = -9x⁷ + 13x⁵ - 4x³ + 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 -9x⁷ because it has the highest exponent (7). Determine the degree.
   Full step-by-step solution

   Step 1: Identify the leading term. The leading term is -9x⁷ because it has the highest exponent (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 -9, which is negative. Step 4: Recall end behavior for odd-degree polynomials: if the leading coefficient is positive, as x → ∞, f(x) → ∞ and as x → -∞, f(x) → -∞. If the leading coefficient is negative, the behavior is reversed: as x → ∞, f(x) → -∞ and as x → -∞, f(x) → ∞. Step 5: Apply to this polynomial: since degree is odd and leading coefficient is negative, as x → ∞, f(x) → -∞ and as x → -∞, f(x) → ∞. The answer is: As x → ∞, f(x) → -∞; as x → -∞, f(x) → ∞.

8. Emma is studying the flight path of a model rocket. The height of the rocket, in meters, after t seconds is modeled by the polynomial function h(t) = -5t⁵ + 25t³ - 10t. Emma wants to describe the long-term behavior of the rocket's height as time increases without bound (t → ∞) and as time goes backward (t → -∞). Determine the end behavior of h(t) using the leading term. Answer: As x → ∞, h(t) → -∞; as x → -∞, h(t) → ∞ Solution: Identify the leading term. The polynomial is h(t) = -5t⁵ + 25t³ - 10t. The term with the highest exponent is -5t⁵.
   Full step-by-step solution

   Step 1: Identify the leading term. The polynomial is h(t) = -5t⁵ + 25t³ - 10t. The term with the highest exponent is -5t⁵. So the leading term is -5t⁵. Step 2: Determine the degree. The exponent of the leading term is 5, which is odd. Step 3: Determine the leading coefficient. The coefficient of the leading term is -5, which is negative. Step 4: Apply end behavior rules. For a polynomial with odd degree and negative leading coefficient: - As t → ∞ (right side), the function goes to -∞. - As t → -∞ (left side), the function goes to ∞. Thus, the end behavior is: as t → ∞, h(t) → -∞; as t → -∞, h(t) → ∞.