Polynomial Analysis

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Polynomial Analysis · Grade 12 · Worksheet 3

1. Matiu is an environmental scientist modeling the population of a rare bird species in a protected forest. The population (in hundreds) over time t (in years) is modeled by the polynomial function P(t) = -2t⁶ + 4t⁴ - 6t² + 10. As time goes on (t → ∞) and as we look back in time (t → -∞), what happens to the bird population? Describe the end behavior of P(t) using the degree and leading coefficient. Answer: As t → ∞, P(t) → -∞; as t → -∞, P(t) → -∞ Solution: Identify the leading term of P(t) = -2t⁶ + 4t⁴ - 6t² + 10. The leading term is -2t⁶, because it has the highest exponent. Determine the degree: the exponent of the leading term is 6, which is an even number.
   Full step-by-step solution

   Step 1: Identify the leading term of P(t) = -2t⁶ + 4t⁴ - 6t² + 10. The leading term is -2t⁶, because it has the highest exponent. Step 2: Determine the degree: the exponent of the leading term is 6, which is an even number. Step 3: Determine the leading coefficient: it is -2, which is negative. Step 4: For a polynomial with an even degree and a negative leading coefficient, both ends of the graph go downward. As t → ∞ (moving forward in time), P(t) → -∞ (population declines without bound). As t → -∞ (looking backward in time), P(t) → -∞ (population also declines without bound). Step 5: Final answer: As t → ∞, P(t) → -∞; as t → -∞, P(t) → -∞.

2. Matiu is analyzing the graph of a polynomial function on a coordinate plane. The graph rises on the left side as x → -∞ and falls on the right side as x → +∞. The function is f(x) = -2x^6 + 4x^4 - 6x^2 + 8. Determine the degree of the polynomial and the sign of its leading coefficient. Answer: degree: 6, leading coefficient: negative Solution: Identify the degree of the polynomial. The highest power of x is 6, so the degree is 6, which is even. Identify the leading coefficient.
   Full step-by-step solution

   Step 1: Identify the degree of the polynomial. The highest power of x is 6, so the degree is 6, which is even. Step 2: Identify the leading coefficient. The leading term is -2x^6, so the leading coefficient is -2, which is negative. Step 3: Recall the end behavior rules for even-degree polynomials: - If leading coefficient is positive: both ends go up (as x→-∞, f(x)→+∞ and as x→+∞, f(x)→+∞) - If leading coefficient is negative: both ends go down (as x→-∞, f(x)→-∞ and as x→+∞, f(x)→-∞) Step 4: For this function, the graph rises on the left (as x→-∞, f(x)→+∞) and falls on the right (as x→+∞, f(x)→-∞). This does not match the typical even-degree pattern where both ends go the same direction. Step 5: Wait — re-read the problem carefully. The problem states the graph rises on the left and falls on the right. This is actually the end behavior of an odd-degree polynomial with a positive leading coefficient, but the given function has even degree 6 with a negative leading coefficient. The described behavior does NOT match the function's actual end behavior. Therefore, the correct answer is based solely on analyzing the function's formula, not the description. Step 6: The degree is 6 (even) and the leading coefficient is -2 (negative). The answer is degree: 6, leading coefficient: negative.

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. f(x) = -3x⁷ + 5x⁴ - 2x + 9. 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 -3x⁷, since it has the highest degree (7). Determine the degree.
   Full step-by-step solution

   Step 1: Identify the leading term. The leading term is -3x⁷, since it has the highest degree (7). Step 2: Determine the degree. The degree is 7, which is odd. Step 3: Determine the leading coefficient. The leading coefficient is -3, which is negative. Step 4: For an odd-degree polynomial with a negative leading coefficient: - As x → ∞ (right side), the function goes to -∞ (downward). - As x → -∞ (left side), the function goes to ∞ (upward). Step 5: Therefore, the end behavior is: as x → ∞, f(x) → -∞; as x → -∞, f(x) → ∞.

5. lim(x→∞) (3x⁴ - 2x³ + 5x - 1)/(4x⁴ + x² - 7) = ? Answer: 3/4 Solution: To find the limit as x approaches infinity of (3x⁴ - 2x³ + 5x - 1)/(4x⁴ + x² - 7), we need to analyze how the numerator and denominator behave when x becomes very large.
   Full step-by-step solution

   To find the limit as x approaches infinity of (3x⁴ - 2x³ + 5x - 1)/(4x⁴ + x² - 7), we need to analyze how the numerator and denominator behave when x becomes very large. Step 1: Identify the dominant terms When x is very large, the highest power term in both numerator and denominator will dominate the behavior of the function. In this case, both have x⁴ as their highest power term. Step 2: Factor out x⁴ from both numerator and denominator Let's factor x⁴ from both the numerator and denominator: Numerator: 3x⁴ - 2x³ + 5x - 1 = x⁴(3 - 2/x + 5/x³ - 1/x⁴) Denominator: 4x⁴ + x² - 7 = x⁴(4 + 1/x² - 7/x⁴) Step 3: Simplify the expression Now we can write our limit as: lim(x→∞) [x⁴(3 - 2/x + 5/x³ - 1/x⁴)] / [x⁴(4 + 1/x² - 7/x⁴)] The x⁴ terms cancel out, giving us: lim(x→∞) (3 - 2/x + 5/x³ - 1/x⁴)/(4 + 1/x² - 7/x⁴) Step 4: Evaluate the limit As x approaches infinity: - 2/x approaches 0 - 5/x³ approaches 0 - 1/x⁴ approaches 0 - 1/x² approaches 0 - 7/x⁴ approaches 0 So our expression becomes: (3 - 0 + 0 - 0)/(4 + 0 - 0) = 3/4 Therefore, the limit as x approaches infinity of (3x⁴ - 2x³ + 5x - 1)/(4x⁴ + x² - 7) is 3/4. This makes sense because when dealing with rational functions where the numerator and denominator have the same highest degree, the limit as x approaches infinity is simply the ratio of the leading coefficients.

6. lim(x→∞) (3x⁴ - 2x³ + 5x - 1)/(2x⁴ + x² - 7) = ? Answer: 3/2 Solution: lim(x→∞) (3x⁴ - 2x³ + 5x - 1) / (2x⁴ + x² - 7) Identify the highest power of x in the denominator The denominator is 2x⁴ + x² - 7. The highest power of x here is x⁴.
   Full step-by-step solution

   Let's find the limit step by step. We want: lim(x→∞) (3x⁴ - 2x³ + 5x - 1) / (2x⁴ + x² - 7) --- **Step 1: Identify the highest power of x in the denominator** The denominator is 2x⁴ + x² - 7. The highest power of x here is x⁴. --- **Step 2: Divide numerator and denominator by x⁴** This is a standard technique for limits at infinity of rational functions. Numerator divided by x⁴: (3x⁴ - 2x³ + 5x - 1) / x⁴ = 3 - 2/x + 5/x³ - 1/x⁴ Denominator divided by x⁴: (2x⁴ + x² - 7) / x⁴ = 2 + 1/x² - 7/x⁴ So the expression becomes: [3 - 2/x + 5/x³ - 1/x⁴] / [2 + 1/x² - 7/x⁴] --- **Step 3: Take the limit as x → ∞** As x → ∞, terms with x in the denominator go to 0: - 2/x → 0 - 5/x³ → 0 - 1/x⁴ → 0 - 1/x² → 0 - 7/x⁴ → 0 So the limit becomes: (3 - 0 + 0 - 0) / (2 + 0 - 0) = 3/2 --- **Step 4: Conclusion** The limit is 3/2. --- ANSWER: 3/2

7. Emma is designing a roller coaster track. The height of the track, in meters above the ground, is modeled by the polynomial function h(x) = -0.02x⁵ + 0.4x³ - 5x + 20, where x represents the horizontal distance from the start in hundreds of meters. Emma wants to describe the end behavior of the track to ensure safety and proper structural support at the far ends. Describe the end behavior of h(x) as x → ∞ and as x → -∞. Answer: As x → ∞, h(x) → -∞; as x → -∞, h(x) → ∞ Solution: Identify the leading term. The polynomial is h(x) = -0.02x⁵ + 0.4x³ - 5x + 20. The leading term is -0.02x⁵, because it has the highest power of x.
   Full step-by-step solution

   Step 1: Identify the leading term. The polynomial is h(x) = -0.02x⁵ + 0.4x³ - 5x + 20. The leading term is -0.02x⁵, because it has the highest power of x. Step 2: Determine the degree. The degree is 5, which is odd. Step 3: Determine the leading coefficient. The leading coefficient is -0.02, which is negative. Step 4: Apply end behavior rules for polynomials: - For an odd-degree polynomial with a negative leading coefficient, as x → ∞, the function goes to -∞, and as x → -∞, the function goes to ∞. Step 5: Write the final description: As x → ∞, h(x) → -∞; as x → -∞, h(x) → ∞.

8. f(x) = 7x⁷ - 2x⁵ + 12x³ - 17. 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) = 7x⁷ - 2x⁵ + 12x³ - 17. The highest power is x⁷, so the leading term is 7x⁷.
   Full step-by-step solution

   Step 1: Identify the leading term. The polynomial is f(x) = 7x⁷ - 2x⁵ + 12x³ - 17. The highest power is x⁷, so the leading term is 7x⁷. Step 2: Determine the degree. The degree is 7, which is odd. Step 3: Determine the leading coefficient. The leading coefficient is 7, which is positive. Step 4: For an odd-degree polynomial with a positive leading coefficient: as x → ∞, f(x) → ∞; as x → -∞, f(x) → -∞. The answer is: As x → ∞, f(x) → ∞; as x → -∞, f(x) → -∞.